{"text_format":"1","workbook":"xlsx","n_sheets":8,"sheets":["OneVarMCSim","TwoVarMCSim","Intro","DGP","MC","MCParr","Optimal","MCParrOpt"],"not_rendered":{"charts":9,"images":2,"macros":1,"external_links":0,"defined_names":2,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"OneVarMCSim","index":0,"n_rows":103,"n_cols":38,"rows_kept":103,"cut":false,"hidden":true,"merged":[[1,5,1,6],[3,8,3,10],[3,11,3,13]]}
["Results of Monte Carlo Simulation",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0]
["Sample Number",null,null,null,null,"Simulation Stats",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,0,0]
[1,null,null,null,null,null,"repetitions",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.12903225421905518,0,null,0.12903225421905518,0]
[2,null,null,null,null,null,"seconds",null,"Summary Statistics",null,null,"Notes",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.25806450843811035,3,null,0.12903225421905518,3]
[3,null,null,null,null,null,null,null,"Average",1.1671042295446972,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.38709676265716553,4,null,0.25806450843811035,3]
[4,null,null,null,null,null,null,null,"SD",0.57603988214435031,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.5161290168762207,8,null,0.25806450843811035,4]
[5,null,null,null,null,null,null,null,"Max",2.5793023004401907,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.64516127109527588,4,null,0.38709676265716553,4]
[6,null,null,null,null,null,null,null,"Min",0.21342784181531316,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.77419352531433105,9,null,0.38709676265716553,8]
[7,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0.90322577953338623,9,null,0.5161290168762207,8]
[8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.0322580337524414,12,null,0.5161290168762207,4]
[9,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.1612902879714966,8,null,0.64516127109527588,4]
[10,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.2903225421905518,3,null,0.64516127109527588,9]
[11,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.4193547964096069,6,null,0.77419352531433105,9]
[12,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.5483870506286621,7,null,0.77419352531433105,9]
[13,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.6774193048477173,6,null,0.90322577953338623,9]
[14,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.8064515590667725,6,null,0.90322577953338623,12]
[15,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1.9354838132858276,2,null,1.0322580337524414,12]
[16,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.0645160675048828,5,null,1.0322580337524414,8]
[17,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.193548321723938,2,null,1.1612902879714966,8]
[18,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.3225805759429932,3,null,1.1612902879714966,3]
[19,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.4516128301620483,2,null,1.2903225421905518,3]
[20,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.5806450843811035,1,null,1.2903225421905518,6]
[21,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.7096773386001587,0,null,1.4193547964096069,6]
[22,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.8387095928192139,0,null,1.4193547964096069,7]
[23,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.967741847038269,0,null,1.5483870506286621,7]
[24,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.0967741012573242,0,null,1.5483870506286621,6]
[25,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.2258063554763794,0,null,1.6774193048477173,6]
[26,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.3548386096954346,0,null,1.6774193048477173,6]
[27,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.4838708639144897,0,null,1.8064515590667725,6]
[28,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.6129031181335449,0,null,1.8064515590667725,2]
[29,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.7419353723526001,0,null,1.9354838132858276,2]
[30,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.8709676265716553,0,null,1.9354838132858276,5]
[31,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.9999998807907104,0,null,2.0645160675048828,5]
[32,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.0645160675048828,2]
[33,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.193548321723938,2]
[34,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.193548321723938,3]
[35,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.3225805759429932,3]
[36,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.3225805759429932,2]
[37,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.4516128301620483,2]
[38,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.4516128301620483,1]
[39,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.5806450843811035,1]
[40,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.5806450843811035,0]
[41,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.7096773386001587,0]
[42,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.7096773386001587,0]
[43,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.8387095928192139,0]
[44,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.8387095928192139,0]
[45,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.967741847038269,0]
[46,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2.967741847038269,0]
[47,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.0967741012573242,0]
[48,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.0967741012573242,0]
[49,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.2258063554763794,0]
[50,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.2258063554763794,0]
[51,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.3548386096954346,0]
[52,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.3548386096954346,0]
[53,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.4838708639144897,0]
[54,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.4838708639144897,0]
[55,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.6129031181335449,0]
[56,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.6129031181335449,0]
[57,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.7419353723526001,0]
[58,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.7419353723526001,0]
[59,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.8709676265716553,0]
[60,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.8709676265716553,0]
[61,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.9999998807907104,0]
[62,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3.9999998807907104,0]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"TwoVarMCSim","index":1,"n_rows":103,"n_cols":13,"rows_kept":103,"cut":false,"hidden":true,"merged":[[1,5,1,6],[3,8,3,9],[3,10,3,11],[3,12,3,13]]}
["Results of Monte Carlo Simulation"]
["Sample Number","Var1","Var2",null,null,"Simulation Stats"]
[1,null,null,null,null,null,"repetitions"]
[2,null,null,null,null,null,"secs",null,"Var1",null,"Var2",null,"Notes"]
[3,null,null,null,null,null,null,null,"Average",3.8919870765226666E-3,"Average",3.3556206172831171E-3]
[4,null,null,null,null,null,null,null,"SD",0.16598652994176216,"SD",0.14882661448398982]
[5,null,null,null,null,null,null,null,"Max",0.50081070155017959,"Max",0.40019025312581391]
[6,null,null,null,null,null,null,null,"Min",-0.54963495662729456,"Min",-0.46734164154276719]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17,null,null,null,null,1]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"Intro","index":2,"n_rows":3,"n_cols":1,"rows_kept":3,"cut":false,"hidden":false,"merged":[]}
["This is a completed version of Parrondo's Paradox."]
[]
["You must ENABLE MACROS for the buttons to work."]

{"sheet":"DGP","index":3,"n_rows":107,"n_cols":27,"rows_kept":107,"cut":false,"hidden":false,"merged":[]}
["Understanding Parrondo's Paradox",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,"Random"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"ABAB"]
["Game A",null,null,null,null,null,"Game B",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"AABB"]
["epsilon",5.0000000000000001E-3,"amount of bias in the coin",null,null,null,"Coin 1",null,null,"Coin 2"]
["Prob Heads",{"$f":"0.5-B4","v":0.495},null,null,null,null,"Prob Heads",{"$f":"0.1-B4","v":9.5000000000000001E-2},null,"Prob Heads",{"$f":"0.75-B4","v":0.745}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Playing both randomly",null,"IF Game B is chosen",null,null,"Game B","Actual"]
["Flip #","Result","End M",null,null,null,"Flip #","Start M","M/3 Remainder","Coin","Result","End M",null,"Flip #","Game","Start M","M/3 Remainder","Coin","Result","Result","End M"]
[1,{"$f":"IF(RAND()<=$B$5,1,0)","v":1},{"$f":"IF(B8=1,1,-1)","v":1},null,null,null,1,0,{"$f":"MOD(H8,3)","v":0},{"$f":"IF(I8=0,1,2)","v":1},{"$f":"IF(I8=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":0},{"$f":"IF(K8=1,1,-1)","v":-1},null,1,{"$f":"IF(RAND()<=0.5,\"A\",\"B\")","v":"B"},0,{"$f":"MOD(P8,3)","v":0},1,{"$f":"IF(RAND()<=$H$5,1,0)","v":0},{"$f":"IF(O8=\"A\",B8,S8)","v":0},{"$f":"IF(T8=1,1,-1)","v":-1}]
[2,0,{"$f":"IF(B9=1,1+C8,-1+C8)","v":0},null,null,null,2,{"$f":"L8","v":-1},2,2,0,{"$f":"IF(K9=1,1+L8,-1+L8)","v":-2},null,2,"A",{"$f":"U8","v":-1},2,{"$f":"IF(Q9=0,1,2)","v":2},{"$f":"IF(Q9=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":1},0,{"$f":"IF(T9=1,1+U8,-1+U8)","v":-2}]
[3,1,1,null,null,null,3,-2,1,2,0,-3,null,3,"A",-2,1,2,1,1,-1]
[4,1,2,null,null,null,4,-3,0,1,0,-4,null,4,"A",-1,2,2,0,1,0]
[5,0,1,null,null,null,5,-4,2,2,1,-3,null,5,"B",0,0,1,0,0,-1]
[6,1,2,null,null,null,6,-3,0,1,0,-4,null,6,"B",-1,2,2,0,0,-2]
[7,0,1,null,null,null,7,-4,2,2,1,-3,null,7,"B",-2,1,2,0,0,-3]
[8,1,2,null,null,null,8,-3,0,1,0,-4,null,8,"A",-3,0,1,0,1,-2]
[9,0,1,null,null,null,9,-4,2,2,1,-3,null,9,"A",-2,1,2,1,0,-3]
[10,1,2,null,null,null,10,-3,0,1,0,-4,null,10,"A",-3,0,1,0,1,-2]
[11,0,1,null,null,null,11,-4,2,2,0,-5,null,11,"B",-2,1,2,1,1,-1]
[12,0,0,null,null,null,12,-5,1,2,1,-4,null,12,"A",-1,2,2,1,0,-2]
[13,0,-1,null,null,null,13,-4,2,2,0,-5,null,13,"A",-2,1,2,0,0,-3]
[14,1,0,null,null,null,14,-5,1,2,1,-4,null,14,"A",-3,0,1,0,1,-2]
[15,0,-1,null,null,null,15,-4,2,2,0,-5,null,15,"B",-2,1,2,0,0,-3]
[16,1,0,null,null,null,16,-5,1,2,1,-4,null,16,"A",-3,0,1,0,1,-2]
[17,1,1,null,null,null,17,-4,2,2,1,-3,null,17,"B",-2,1,2,1,1,-1]
[18,1,2,null,null,null,18,-3,0,1,0,-4,null,18,"B",-1,2,2,1,1,0]
[19,1,3,null,null,null,19,-4,2,2,1,-3,null,19,"B",0,0,1,0,0,-1]
[20,1,4,null,null,null,20,-3,0,1,0,-4,null,20,"B",-1,2,2,1,1,0]
[21,1,5,null,null,null,21,-4,2,2,1,-3,null,21,"A",0,0,1,0,1,1]
[22,0,4,null,null,null,22,-3,0,1,0,-4,null,22,"B",1,1,2,0,0,0]
[23,0,3,null,null,null,23,-4,2,2,0,-5,null,23,"B",0,0,1,0,0,-1]
[24,0,2,null,null,null,24,-5,1,2,1,-4,null,24,"B",-1,2,2,1,1,0]
[25,0,1,null,null,null,25,-4,2,2,1,-3,null,25,"B",0,0,1,0,0,-1]
[26,1,2,null,null,null,26,-3,0,1,0,-4,null,26,"A",-1,2,2,1,1,0]
[27,0,1,null,null,null,27,-4,2,2,1,-3,null,27,"A",0,0,1,0,0,-1]
[28,1,2,null,null,null,28,-3,0,1,0,-4,null,28,"B",-1,2,2,1,1,0]
[29,1,3,null,null,null,29,-4,2,2,0,-5,null,29,"A",0,0,1,0,1,1]
[30,0,2,null,null,null,30,-5,1,2,0,-6,null,30,"A",1,1,2,1,0,0]
[31,1,3,null,null,null,31,-6,0,1,0,-7,null,31,"B",0,0,1,0,0,-1]
[32,1,4,null,null,null,32,-7,2,2,1,-6,null,32,"A",-1,2,2,0,1,0]
[33,{"$f":"IF(RAND()<=$B$5,1,0)","v":0},3,null,null,null,33,-6,{"$f":"MOD(H40,3)","v":0},{"$f":"IF(I40=0,1,2)","v":1},{"$f":"IF(I40=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":0},-7,null,33,{"$f":"IF(RAND()<=0.5,\"A\",\"B\")","v":"B"},0,{"$f":"MOD(P40,3)","v":0},1,0,{"$f":"IF(O40=\"A\",B40,S40)","v":0},-1]
[34,1,{"$f":"IF(B41=1,1+C40,-1+C40)","v":4},null,null,null,34,{"$f":"L40","v":-7},2,2,1,{"$f":"IF(K41=1,1+L40,-1+L40)","v":-6},null,34,"A",{"$f":"U40","v":-1},2,{"$f":"IF(Q41=0,1,2)","v":2},{"$f":"IF(Q41=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":1},1,{"$f":"IF(T41=1,1+U40,-1+U40)","v":0}]
[35,0,3,null,null,null,35,-6,0,1,0,-7,null,35,"A",0,0,1,0,0,-1]
[36,1,4,null,null,null,36,-7,2,2,1,-6,null,36,"B",-1,2,2,1,1,0]
[37,1,5,null,null,null,37,-6,0,1,0,-7,null,37,"B",0,0,1,0,0,-1]
[38,1,6,null,null,null,38,-7,2,2,1,-6,null,38,"B",-1,2,2,0,0,-2]
[39,1,7,null,null,null,39,-6,0,1,0,-7,null,39,"B",-2,1,2,1,1,-1]
[40,1,8,null,null,null,40,-7,2,2,1,-6,null,40,"B",-1,2,2,0,0,-2]
[41,1,9,null,null,null,41,-6,0,1,0,-7,null,41,"B",-2,1,2,1,1,-1]
[42,1,10,null,null,null,42,-7,2,2,0,-8,null,42,"A",-1,2,2,1,1,0]
[43,1,11,null,null,null,43,-8,1,2,0,-9,null,43,"A",0,0,1,0,1,1]
[44,0,10,null,null,null,44,-9,0,1,0,-10,null,44,"A",1,1,2,1,0,0]
[45,0,9,null,null,null,45,-10,2,2,1,-9,null,45,"A",0,0,1,0,0,-1]
[46,1,10,null,null,null,46,-9,0,1,0,-10,null,46,"A",-1,2,2,1,1,0]
[47,0,9,null,null,null,47,-10,2,2,1,-9,null,47,"B",0,0,1,0,0,-1]
[48,0,8,null,null,null,48,-9,0,1,0,-10,null,48,"B",-1,2,2,1,1,0]
[49,0,7,null,null,null,49,-10,2,2,0,-11,null,49,"A",0,0,1,0,0,-1]
[50,0,6,null,null,null,50,-11,1,2,1,-10,null,50,"A",-1,2,2,1,0,-2]
[51,1,7,null,null,null,51,-10,2,2,1,-9,null,51,"A",-2,1,2,0,1,-1]
[52,1,8,null,null,null,52,-9,0,1,0,-10,null,52,"B",-1,2,2,1,1,0]
[53,0,7,null,null,null,53,-10,2,2,0,-11,null,53,"B",0,0,1,0,0,-1]
[54,0,6,null,null,null,54,-11,1,2,0,-12,null,54,"B",-1,2,2,1,1,0]
[55,0,5,null,null,null,55,-12,0,1,0,-13,null,55,"A",0,0,1,0,0,-1]
[56,1,6,null,null,null,56,-13,2,2,1,-12,null,56,"B",-1,2,2,0,0,-2]
[57,1,7,null,null,null,57,-12,0,1,0,-13,null,57,"B",-2,1,2,1,1,-1]
[58,0,6,null,null,null,58,-13,2,2,1,-12,null,58,"B",-1,2,2,1,1,0]
[59,1,7,null,null,null,59,-12,0,1,0,-13,null,59,"B",0,0,1,1,1,1]
[60,0,6,null,null,null,60,-13,2,2,1,-12,null,60,"B",1,1,2,1,1,2]
[61,0,5,null,null,null,61,-12,0,1,0,-13,null,61,"A",2,2,2,1,0,1]
[62,0,4,null,null,null,62,-13,2,2,1,-12,null,62,"A",1,1,2,1,0,0]
[63,1,5,null,null,null,63,-12,0,1,0,-13,null,63,"B",0,0,1,0,0,-1]
[64,1,6,null,null,null,64,-13,2,2,0,-14,null,64,"B",-1,2,2,0,0,-2]
[65,{"$f":"IF(RAND()<=$B$5,1,0)","v":1},7,null,null,null,65,-14,{"$f":"MOD(H72,3)","v":1},{"$f":"IF(I72=0,1,2)","v":2},{"$f":"IF(I72=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":0},-15,null,65,{"$f":"IF(RAND()<=0.5,\"A\",\"B\")","v":"A"},-2,{"$f":"MOD(P72,3)","v":1},2,1,{"$f":"IF(O72=\"A\",B72,S72)","v":1},-1]
[66,1,{"$f":"IF(B73=1,1+C72,-1+C72)","v":8},null,null,null,66,{"$f":"L72","v":-15},0,1,0,{"$f":"IF(K73=1,1+L72,-1+L72)","v":-16},null,66,"B",{"$f":"U72","v":-1},2,{"$f":"IF(Q73=0,1,2)","v":2},{"$f":"IF(Q73=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":0},0,{"$f":"IF(T73=1,1+U72,-1+U72)","v":-2}]
[67,0,7,null,null,null,67,-16,2,2,0,-17,null,67,"B",-2,1,2,1,1,-1]
[68,0,6,null,null,null,68,-17,1,2,1,-16,null,68,"B",-1,2,2,0,0,-2]
[69,0,5,null,null,null,69,-16,2,2,1,-15,null,69,"A",-2,1,2,1,0,-3]
[70,1,6,null,null,null,70,-15,0,1,0,-16,null,70,"A",-3,0,1,0,1,-2]
[71,1,7,null,null,null,71,-16,2,2,0,-17,null,71,"A",-2,1,2,1,1,-1]
[72,0,6,null,null,null,72,-17,1,2,0,-18,null,72,"B",-1,2,2,1,1,0]
[73,0,5,null,null,null,73,-18,0,1,0,-19,null,73,"A",0,0,1,0,0,-1]
[74,0,4,null,null,null,74,-19,2,2,1,-18,null,74,"B",-1,2,2,1,1,0]
[75,1,5,null,null,null,75,-18,0,1,0,-19,null,75,"B",0,0,1,0,0,-1]
[76,1,6,null,null,null,76,-19,2,2,1,-18,null,76,"A",-1,2,2,1,1,0]
[77,0,5,null,null,null,77,-18,0,1,0,-19,null,77,"A",0,0,1,1,0,-1]
[78,0,4,null,null,null,78,-19,2,2,0,-20,null,78,"A",-1,2,2,1,0,-2]
[79,1,5,null,null,null,79,-20,1,2,1,-19,null,79,"B",-2,1,2,1,1,-1]
[80,1,6,null,null,null,80,-19,2,2,1,-18,null,80,"A",-1,2,2,1,1,0]
[81,0,5,null,null,null,81,-18,0,1,0,-19,null,81,"B",0,0,1,0,0,-1]
[82,1,6,null,null,null,82,-19,2,2,1,-18,null,82,"A",-1,2,2,1,1,0]
[83,1,7,null,null,null,83,-18,0,1,0,-19,null,83,"B",0,0,1,0,0,-1]
[84,1,8,null,null,null,84,-19,2,2,1,-18,null,84,"A",-1,2,2,1,1,0]
[85,0,7,null,null,null,85,-18,0,1,0,-19,null,85,"B",0,0,1,0,0,-1]
[86,1,8,null,null,null,86,-19,2,2,1,-18,null,86,"B",-1,2,2,0,0,-2]
[87,1,9,null,null,null,87,-18,0,1,0,-19,null,87,"B",-2,1,2,1,1,-1]
[88,1,10,null,null,null,88,-19,2,2,1,-18,null,88,"B",-1,2,2,1,1,0]
[89,1,11,null,null,null,89,-18,0,1,0,-19,null,89,"B",0,0,1,0,0,-1]
[90,1,12,null,null,null,90,-19,2,2,0,-20,null,90,"B",-1,2,2,1,1,0]
[91,0,11,null,null,null,91,-20,1,2,1,-19,null,91,"B",0,0,1,0,0,-1]
[92,0,10,null,null,null,92,-19,2,2,0,-20,null,92,"B",-1,2,2,0,0,-2]
[93,1,11,null,null,null,93,-20,1,2,1,-19,null,93,"B",-2,1,2,0,0,-3]
[94,0,10,null,null,null,94,-19,2,2,1,-18,null,94,"A",-3,0,1,1,0,-4]
[95,1,11,null,null,null,95,-18,0,1,0,-19,null,95,"A",-4,2,2,0,1,-3]
[96,0,10,null,null,null,96,-19,2,2,0,-20,null,96,"A",-3,0,1,0,0,-4]
[97,1,11,null,null,null,97,-20,{"$f":"MOD(H104,3)","v":1},{"$f":"IF(I104=0,1,2)","v":2},0,-21,null,97,"A",-4,{"$f":"MOD(P104,3)","v":2},2,1,{"$f":"IF(O104=\"A\",B104,S104)","v":1},-3]
[98,0,{"$f":"IF(B105=1,1+C104,-1+C104)","v":10},null,null,null,98,-21,{"$f":"MOD(H105,3)","v":0},{"$f":"IF(I105=0,1,2)","v":1},0,{"$f":"IF(K105=1,1+L104,-1+L104)","v":-22},null,98,"B",-3,{"$f":"MOD(P105,3)","v":0},{"$f":"IF(Q105=0,1,2)","v":1},0,{"$f":"IF(O105=\"A\",B105,S105)","v":0},{"$f":"IF(T105=1,1+U104,-1+U104)","v":-4}]
[99,0,{"$f":"IF(B106=1,1+C105,-1+C105)","v":9},null,null,null,99,-22,{"$f":"MOD(H106,3)","v":2},{"$f":"IF(I106=0,1,2)","v":2},1,{"$f":"IF(K106=1,1+L105,-1+L105)","v":-21},null,99,"A",-4,{"$f":"MOD(P106,3)","v":2},{"$f":"IF(Q106=0,1,2)","v":2},1,{"$f":"IF(O106=\"A\",B106,S106)","v":0},{"$f":"IF(T106=1,1+U105,-1+U105)","v":-5}]
[100,0,{"$f":"IF(B107=1,1+C106,-1+C106)","v":8},null,null,null,100,-21,{"$f":"MOD(H107,3)","v":0},{"$f":"IF(I107=0,1,2)","v":1},0,{"$f":"IF(K107=1,1+L106,-1+L106)","v":-22},null,100,"A",-5,{"$f":"MOD(P107,3)","v":1},{"$f":"IF(Q107=0,1,2)","v":2},1,{"$f":"IF(O107=\"A\",B107,S107)","v":0},{"$f":"IF(T107=1,1+U106,-1+U106)","v":-6}]

{"sheet":"MC","index":4,"n_rows":141,"n_cols":11,"rows_kept":141,"cut":false,"hidden":true,"merged":[[2,4,2,6]]}
["Analytical Solution via Markov Chain"]
[null,null,null,null,null,null,null,null,"Initial M",0]
["epsilon",5.0000000000000001E-3,null,null,"Mod(M,3)"]
["Transition Matrix: Game B",null,null,null,0,1,2,null,"Flip #","Expected M"]
[0,{"$f":"0.1-B3","v":9.5000000000000001E-2},{"$f":"0.9+B3","v":0.90500000000000003},null,{"$f":"IF(MOD($J$2,3)=0,1,0)","v":1},{"$f":"IF(MOD($J$2,3)=1,1,0)","v":0},{"$f":"IF(MOD($J$2,3)=2,1,0)","v":0},null,1,{"$f":"E5*($B$5-$C$5)+F5*($C$6-$A$6)+G5*($A$7-$B$7)+J2","v":-0.81}]
[{"$f":"0.25+B3","v":0.255},0,{"$f":"0.75-B3","v":0.745},null,{"$f":"MMULT(E5:G5,$A$5:$C$7)","v":0},9.5000000000000001E-2,0.90500000000000003,null,2,{"$f":"E6*($B$5-$C$5)+F6*($C$6-$A$6)+G6*($A$7-$B$7)+J5","v":-0.32000000000000006}]
[{"$f":"0.75-B3","v":0.745},{"$f":"0.25+B3","v":0.255},0,null,{"$f":"MMULT(E6:G6,$A$5:$C$7)","v":0.69845000000000002},0.23077500000000001,7.0775000000000005E-2,null,3,-0.73798500000000011]
[null,null,null,null,{"$f":"MMULT(E7:G7,$A$5:$C$7)","v":0.11157500000000001},8.4400375E-2,0.80402462500000005,null,4,-0.39303250000000012]
[null,null,null,null,{"$f":"MMULT(E8:G8,$A$5:$C$7)","v":0.62052044124999994},0.21562590437500001,0.16385365437499999,null,5,-0.70970907362500013]
[null,null,null,null,{"$f":"MMULT(E9:G9,$A$5:$C$7)","v":0.177055578125},0.10073212378437499,0.72221229809062493,null,6,-0.44988132518750018]
[null,null,null,null,{"$f":"MMULT(E10:G10,$A$5:$C$7)","v":0.56373485364253118},0.20098441593498434,0.23528073042248437,null,7,-0.69273663492279081]
[null,null,null,null,{"$f":"MMULT(E11:G11,$A$5:$C$7)","v":0.22653517022817188},0.11355139735377398,0.65991343241805411,null,8,-0.49723235621941425]
[null,null,null,null,{"$f":"MMULT(E12:G12,$A$5:$C$7)","v":0.52059111347666265},0.18979876643828011,0.28961012008505715,null,9,-0.68400080373907568]
[null,null,null,null,{"$f":"MMULT(E13:G13,$A$5:$C$7)","v":0.26415822490512897},0.12330673640197252,0.61253503869289838,null,10,-0.53740649611574343]
[null,null,null,null,{"$f":"MMULT(E14:G14,$A$5:$C$7)","v":0.48778182160871225},0.18129146623267633,0.33092671215861125,null,11,-0.68152286420706942]
[null,null,null,null,{"$f":"MMULT(E15:G15,$A$5:$C$7)","v":0.29276972444749783},0.13072558465327352,0.57650469089922851,null,12,-0.57212350598881667]
[null,null,null,null,{"$f":"MMULT(E16:G16,$A$5:$C$7)","v":0.46283101880650995},0.17482182000181556,0.36234716119167432,null,13,-0.68380383043727966]
[null,null,null,null,{"$f":"MMULT(E17:G17,$A$5:$C$7)","v":0.31452819918826036},0.13636747289049539,0.54910432792124408,null,14,-0.60269048938201819]
[null,null,null,null,{"$f":"MMULT(E18:G18,$A$5:$C$7)","v":0.44385642988840318},0.16990178254280197,0.38624178756879468,null,15,-0.68970384823694242]
[null,null,null,null,{"$f":"MMULT(E19:G19,$A$5:$C$7)","v":0.33107508628716653},0.14065801666944094,0.52826689704339236,null,16,-0.63010146041025905]
[null,null,null,null,{"$f":"MMULT(E20:G20,$A$5:$C$7)","v":0.42942663254803476},0.16616019194334589,0.40441317550861922,null,17,-0.69835608272270433]
[null,null,null,null,{"$f":"MMULT(E21:G21,$A$5:$C$7)","v":0.34365866469947454},0.14392088984676121,0.5124204454537642,null,18,-0.65511234683202124]
[null,null,null,null,{"$f":"MMULT(E22:G22,$A$5:$C$7)","v":0.41845305877397843},0.16331478673715996,0.41823215448886158,null,19,-0.70910132323819319]
[null,null,null,null,{"$f":"MMULT(E23:G23,$A$5:$C$7)","v":0.35322822571217766},0.14640223997818766,0.5003695343096346,null,20,-0.67829801666402423]
[null,null,null,null,{"$f":"MMULT(E24:G24,$A$5:$C$7)","v":0.41010787425511563},0.16115091269161369,0.42874121305327056,null,21,-0.72143825319567467]
[null,null,null,null,{"$f":"MMULT(E25:G25,$A$5:$C$7)","v":0.36050568646104808},0.14828925738281998,0.49120505615613186,null,22,-0.70009564559503723]
[null,null,null,null,{"$f":"MMULT(E26:G26,$A$5:$C$7)","v":0.40376152746893734},0.15950532953361318,0.43673314299744936,null,23,-0.73498563130465588]
[null,null,null,null,{"$f":"MMULT(E27:G27,$A$5:$C$7)","v":0.3660400505641711},0.14972429657389863,0.48423565286193015,null,24,-0.72083769703807843]
[null,null,null,null,{"$f":"MMULT(E28:G28,$A$5:$C$7)","v":0.39893525700848215},0.15825389628338846,0.44281084670812931,null,25,-0.74945353114910529]
[null,null,null,null,{"$f":"MMULT(E29:G29,$A$5:$C$7)","v":0.37024882434982037},0.15081561532637877,0.47893556032380075,null,26,-0.74077700280387182]
[null,null,null,null,{"$f":"MMULT(E30:G30,$A$5:$C$7)","v":0.39526497434945812},0.15730220619580212,0.44743281945473962,null,27,-0.76462146945816745]
[null,null,null,null,{"$f":"MMULT(E31:G31,$A$5:$C$7)","v":0.37344951307371055},0.15164554152415713,0.47490494540213218,null,28,-0.76010583645399121]
[null,null,null,null,{"$f":"MMULT(E32:G32,$A$5:$C$7)","v":0.39247379741324856},0.15657846481954621,0.45094773776720509,null,29,-0.78032177309121442]
[null,null,null,null,{"$f":"MMULT(E33:G33,$A$5:$C$7)","v":0.3758835731655521},0.1522766838848959,0.47183974294955189,null,30,-0.7789704182064322]
[null,null,null,null,{"$f":"MMULT(E34:G34,$A$5:$C$7)","v":0.39035116288806465},0.15602807390286319,0.45362076320907208,null,31,-0.79642692996091635]
[null,null,null,null,{"$f":"MMULT(E35:G35,$A$5:$C$7)","v":0.37773462743598879},0.15275665509267952,0.4695087174713316,null,32,-0.7974819456277018]
[null,null,null,null,{"$f":"MMULT(E36:G36,$A$5:$C$7)","v":0.38873694156477534},0.15560951256160849,0.45565354587361606,null,33,-0.81283996966190974]
[null,null,null,null,{"$f":"MMULT(E37:G37,$A$5:$C$7)","v":0.37914231737905413},0.15312166364642577,0.46773601897452,null,34,{"$f":"E38*($B$5-$C$5)+F38*($C$6-$A$6)+G38*($A$7-$B$7)+J37","v":-0.81572498225468026}]
[null,null,null,null,{"$f":"MMULT(E38:G38,$A$5:$C$7)","v":0.38750935836585593},0.15529120498951274,0.45719943664463114,null,35,-0.82948714813029312]
[null,null,null,null,{"$f":"MMULT(E39:G39,$A$5:$C$7)","v":0.38021283757257596},0.15339924538913724,0.46638791703828664,null,36,-0.83376383697464196]
[null,null,null,null,{"$f":"MMULT(E40:G40,$A$5:$C$7)","v":0.38657580576775352},0.15504913841415782,0.45837505581808846,null,37,-0.84631238447272161]
[null,null,null,null,{"$f":"MMULT(E41:G41,$A$5:$C$7)","v":0.38102694688008615},0.15361034078154914,0.46536271233836457,null,38,-0.85164741541683364]
[null,null,null,null,{"$f":"MMULT(E42:G42,$A$5:$C$7)","v":0.38586585759137659},0.15486505159989117,0.45926909080873207,null,39,-0.8632730302856233]
[null,null,null,null,{"$f":"MMULT(E43:G43,$A$5:$C$7)","v":0.38164606081047764},0.15377087462740746,0.46458306456211473,null,40,-0.86941290933924431]
[null,null,null,null,{"$f":"MMULT(E44:G44,$A$5:$C$7)","v":0.38532595612876436},0.15472505724033464,0.45994898663090084,null,41,-0.88033665230663805]
[null,null,null,null,{"$f":"MMULT(E45:G45,$A$5:$C$7)","v":0.38211688463630644},0.15389295742311232,0.46399015794058107,null,42,-0.88708860233383657]
[null,null,null,null,{"$f":"MMULT(E46:G46,$A$5:$C$7)","v":0.38491537180862656},0.15461859431529729,0.46046603387607599,null,43,-0.89747858568505112]
[null,null,null,null,{"$f":"MMULT(E47:G47,$A$5:$C$7)","v":0.38247493678807742},0.15398579896021891,0.46353926425170355,null,44,-0.90469600350955182]
[null,null,null,null,{"$f":"MMULT(E48:G48,$A$5:$C$7)","v":0.384603130602375},0.15453763137905174,0.46085923801857315,null,45,-0.91468007329263945]
[null,null,null,null,{"$f":"MMULT(E49:G49,$A$5:$C$7)","v":0.38274722832549518},0.15405640310196178,0.46319636857254293,null,46,-0.92225147011578334]
[null,null,null,null,{"$f":"MMULT(E50:G50,$A$5:$C$7)","v":0.38436567737754473},0.15447606067692049,0.46115826194553466,null,47,-0.93192685070659154]
[null,null,null,null,{"$f":"MMULT(E51:G51,$A$5:$C$7)","v":0.38295430062203806},0.1541100961469781,0.46293560323098376,null,48,-0.93976744151524105]
[null,null,null,null,{"$f":"MMULT(E52:G52,$A$5:$C$7)","v":0.38418509892456232},0.15442923738299447,0.46138566369244316,null,49,-0.94920807011717212]
[null,null,null,null,{"$f":"MMULT(E53:G53,$A$5:$C$7)","v":0.38311177498353371},0.15415092863940644,0.46273729637705979,null,50,-0.95725337759576601]
[null,null,null,null,{"$f":"MMULT(E54:G54,$A$5:$C$7)","v":0.38404777260395823},0.15439362919958596,0.46155859819645578,null,51,-0.96651548198091175]
[null,null,null,null,{"$f":"MMULT(E55:G55,$A$5:$C$7)","v":0.383231531102254},0.15418198093747226,0.46258648796027374,null,52,-0.97471647241384196]
[null,null,null,null,{"$f":"MMULT(E56:G56,$A$5:$C$7)","v":0.38394333866945934},0.15436654988458393,0.46169011144595673,null,53,-0.9838428126841392]
[null,null,null,null,{"$f":"MMULT(E57:G57,$A$5:$C$7)","v":0.38332260324780665},0.15420559559231761,0.46247180115987574,null,54,-0.99216219690628782]
[null,null,null,null,{"$f":"MMULT(E58:G58,$A$5:$C$7)","v":0.38386391874014841},0.15434595660430994,0.46179012465554165,null,55,-1.0011852912684809]
[null,null,null,null,{"$f":"MMULT(E59:G59,$A$5:$C$7)","v":0.38339186180247753},0.15422355406747723,0.46238458413004524,null,56,-1.0095947116117017]
[null,null,null,null,{"$f":"MMULT(E60:G60,$A$5:$C$7)","v":0.38380352146409041},0.15433029582439689,0.4618661827115127,null,57,-1.0185392895150192]
[null,null,null,null,{"$f":"MMULT(E61:G61,$A$5:$C$7)","v":0.38344453155529817},0.15423721113052433,0.46231825731417753,null,58,-1.0270171805369068]
[null,null,null,null,{"$f":"MMULT(E62:G62,$A$5:$C$7)","v":0.38375759053734598},0.15431838611286861,0.4619240233497855,null,59,-1.0359020482354566]
[null,null,null,null,{"$f":"MMULT(E63:G63,$A$5:$C$7)","v":0.38348458585437167},0.15424759705524316,0.46226781709038522,null,60,-1.0444320098461399]
[null,null,null,null,{"$f":"MMULT(E64:G64,$A$5:$C$7)","v":0.38372266098142399},0.15430932901421354,0.46196801000436255,null,61,-1.0532714691219911]
[null,null,null,null,{"$f":"MMULT(E65:G65,$A$5:$C$7)","v":0.3835150463518745},0.15425549534434774,0.46222945830377782,null,62,-1.061841029379428]
[null,null,null,null,{"$f":"MMULT(E66:G66,$A$5:$C$7)","v":0.38369609774912317},0.15430244127089143,0.46200146097998551,null,63,-1.0706459564532882]
[null,null,null,null,{"$f":"MMULT(E67:G67,$A$5:$C$7)","v":0.38353821095416651},0.15426150183606302,0.46220028720977058,null,64,-1.0792456306937046]
[null,null,null,null,{"$f":"MMULT(E68:G68,$A$5:$C$7)","v":0.38367589693947518},0.15429720327913732,0.46202689978138761,null,65,-1.0880242967150222]
[null,null,null,null,{"$f":"MMULT(E69:G69,$A$5:$C$7)","v":0.38355582717331377},0.15426606965350398,0.46217810317318236,null,66,{"$f":"E70*($B$5-$C$5)+F70*($C$6-$A$6)+G70*($A$7-$B$7)+J69","v":-1.09664687204033}]
[null,null,null,null,{"$f":"MMULT(E70:G70,$A$5:$C$7)","v":0.38366053462566435},0.15429321989062633,0.46204624548370943,null,67,-1.1054055670536935]
[null,null,null,null,{"$f":"MMULT(E71:G71,$A$5:$C$7)","v":0.38356922395747323},0.15426954338778404,0.46216123265474285,null,68,-1.1140455581984088]
[null,null,null,null,{"$f":"MMULT(E72:G72,$A$5:$C$7)","v":0.38364885189166836},0.15429019060291937,0.46206095750541243,null,69,-1.1227890656575776]
[null,null,null,null,{"$f":"MMULT(E73:G73,$A$5:$C$7)","v":0.38357941194527667},0.15427218509358867,0.4621484029611348,null,70,-1.1314423011864372]
[null,null,null,null,{"$f":"MMULT(E74:G74,$A$5:$C$7)","v":0.38363996740491058},0.15428788688989065,0.46207214570519894,null,71,-1.140174258812821]
[null,null,null,null,{"$f":"MMULT(E75:G75,$A$5:$C$7)","v":0.38358715970729534},0.15427419405829224,0.46213864623441259,null,72,-1.1488375664323049]
[null,null,null,null,{"$f":"MMULT(E76:G76,$A$5:$C$7)","v":0.38363321092950192},0.15428613496196827,0.46208065410853,null,73,-1.1575607406406574]
[null,null,null,null,{"$f":"MMULT(E77:G77,$A$5:$C$7)","v":0.38359305172615676},0.15427572183597782,0.46213122643786564,null,74,-1.1662317078846611]
[null,null,null,null,{"$f":"MMULT(E78:G78,$A$5:$C$7)","v":0.38362807276438426},0.15428480265564065,0.46208712457997536,null,75,-1.1749482024783606]
[null,null,null,null,{"$f":"MMULT(E79:G79,$A$5:$C$7)","v":0.38359753248926998},0.15427688368051024,0.46212558383022007,null,76,-1.1836249947144115]
[null,null,null,null,{"$f":"MMULT(E80:G80,$A$5:$C$7)","v":0.38362416529204407},0.15428378946318677,0.46209204524476943,null,77,-1.1923364095940687]
[null,null,null,null,{"$f":"MMULT(E81:G81,$A$5:$C$7)","v":0.38360094002046585},0.1542777672401604,0.46212129273937408,null,78,-1.2010176316206742]
[null,null,null,null,{"$f":"MMULT(E82:G82,$A$5:$C$7)","v":0.3836211937370746},0.15428301895048466,0.46209578731244111,null,79,-1.2097251834788709]
[null,null,null,null,{"$f":"MMULT(E83:G83,$A$5:$C$7)","v":0.38360353138014219},0.15427843916969458,0.46211802945016361,null,80,-1.2184097742730555]
[null,null,null,null,{"$f":"MMULT(E84:G84,$A$5:$C$7)","v":0.38361893392864399},0.15428243299090524,0.46209863308045118,null,81,-1.2271143883802926]
[null,null,null,null,{"$f":"MMULT(E85:G85,$A$5:$C$7)","v":0.38360550205761701},0.15427895015873622,0.46211554778364722,null,82,-1.2358015410551946]
[null,null,null,null,{"$f":"MMULT(E86:G86,$A$5:$C$7)","v":0.38361721538929494},0.15428198738030366,0.46210079723040187,null,83,-1.2445039210612778]
[null,null,null,null,{"$f":"MMULT(E87:G87,$A$5:$C$7)","v":0.38360700071862686},0.15427933875573552,0.46211366052563818,null,84,-1.2531930219954925]
[null,null,null,null,{"$f":"MMULT(E88:G88,$A$5:$C$7)","v":0.38361590847431298},0.1542816485023073,0.46210244302338027,null,85,-1.2618937030120991]
[null,null,null,null,{"$f":"MMULT(E89:G89,$A$5:$C$7)","v":0.38360814042050667},0.1542796342760217,0.46211222530347218,null,86,-1.2705842855587575]
[null,null,null,null,{"$f":"MMULT(E90:G90,$A$5:$C$7)","v":0.38361491459147234},0.15428139079233355,0.46210369461619472,null,87,-1.2792836745276712]
[null,null,null,null,{"$f":"MMULT(E91:G91,$A$5:$C$7)","v":0.38360900714111007},0.15427985901331953,0.46211113384557101,null,88,-1.2879753838111141]
[null,null,null,null,{"$f":"MMULT(E92:G92,$A$5:$C$7)","v":0.38361415876334692},0.15428119480902608,0.46210464642762766,null,89,-1.2966737902034648]
[null,null,null,null,{"$f":"MMULT(E93:G93,$A$5:$C$7)","v":0.38360966626488424},0.15428002992156301,0.46211030381355339,null,90,-1.3053663563478142]
[null,null,null,null,{"$f":"MMULT(E94:G94,$A$5:$C$7)","v":0.3836135839710958},0.15428104576762011,0.46210537026128468,null,91,-1.3140640155102385]
[null,null,null,null,{"$f":"MMULT(E95:G95,$A$5:$C$7)","v":0.38361016751540022},0.15428015989388169,0.46210967259071867,null,92,-1.3227572332802586]
[null,null,null,null,{"$f":"MMULT(E96:G96,$A$5:$C$7)","v":0.38361314685302522},0.15428093242459628,0.4621059207223791,null,93,-1.3314543241891912]
[null,null,null,null,{"$f":"MMULT(E97:G97,$A$5:$C$7)","v":0.38361054870644445},0.15428025873524409,0.46210919255831207,null,94,-1.3401480375075687]
[null,null,null,null,{"$f":"MMULT(E98:G98,$A$5:$C$7)","v":0.38361281443342971},0.15428084622948179,0.46210633933708911,null,95,-1.348844696271027]
[null,null,null,null,{"$f":"MMULT(E99:G99,$A$5:$C$7)","v":0.38361083859464923},0.15428033390213355,0.46210882750321786,null,96,-1.3575387864440707]
[null,null,null,null,{"$f":"MMULT(E100:G100,$A$5:$C$7)","v":0.38361256163494134},0.15428078067981224,0.46210665768524706,null,97,-1.3662351165694941]
[null,null,null,null,{"$f":"MMULT(E101:G101,$A$5:$C$7)","v":0.38361105904886117},0.15428039106505742,0.46210854988608202,null,98,-1.3749294933330134]
[null,null,null,null,{"$f":"MMULT(E102:G102,$A$5:$C$7)","v":0.38361236938672072},0.15428073083059274,0.46210689978268715,null,99,-1.3836255735357501]
[null,null,null,null,{"$f":"MMULT(E103:G103,$A$5:$C$7)","v":0.38361122669990305},0.1542804345363237,0.46210833876377388,null,100,-1.3923201682456239,"Final M"]
[]
[]
[null,null,null,null,"Finding Exact Steady-state probabilities (as # flips --> infinity)"]
[null,null,null,null,"Transpose the transition matrix:"]
[null,null,null,null,{"$f":"TRANSPOSE(A5:C7)","v":0},0.255,0.745]
[null,null,null,null,9.5000000000000001E-2,0,0.255]
[null,null,null,null,0.90500000000000003,0.745,0]
[]
[null,null,null,null,"Subtract the identity matrix:"]
[null,null,null,null,-1,{"$f":"F109","v":0.255},{"$f":"G109","v":0.745}]
[null,null,null,null,{"$f":"E110","v":9.5000000000000001E-2},-1,{"$f":"G110","v":0.255}]
[null,null,null,null,{"$f":"E111","v":0.90500000000000003},{"$f":"F111","v":0.745},-1]
[]
[null,null,null,null,"Replace the last row (any row would work) with the "]
[null,null,null,null,"constraint that the probabilities sum to one:"]
[null,null,null,null,{"$f":"E114","v":-1},0.255,0.745,"p0",null,0]
[null,null,null,null,9.5000000000000001E-2,-1,0.255,"p1","=",0]
[null,null,null,null,1,1,1,"p2",null,1]
[]
[null,null,null,null,"Invert the matrix:"]
[null,null,null,null,{"$f":"MINVERSE(E120:G122)","v":-0.59434308513787104},0.23205427228490577,0.38361175899506289]
[null,null,null,null,7.5772823603234646E-2,-0.82639735742277687,0.15428057255839836]
[null,null,null,null,0.51857026153463637,0.59434308513787093,0.46210766844653867]
[]
[null,null,null,null,"Multiply both sides by the inverse to find the steady-state solution:"]
[null,null,null,null,{"$f":"MMULT(E125:G127,J120:J122)","v":0.38361175899506289}]
[null,null,null,null,0.15428057255839836]
[null,null,null,null,0.46210766844653867]
[]
[null,null,null,null,"Difference between probabilties at 100th flip and steady-state:"]
[null,null,null,null,"Position","100th flip","Steady-state","Difference","Conclusion"]
[null,null,null,null,0,{"$f":"E104","v":0.38361122669990305},{"$f":"E130","v":0.38361175899506289},{"$f":"F136-G136","v":-5.3229515983854725E-7},"After 100 flips, the probabilities"]
[null,null,null,null,1,{"$f":"F104","v":0.1542804345363237},{"$f":"E131","v":0.15428057255839836},{"$f":"F137-G137","v":-1.3802207465829497E-7},{"$f":"\"are\"&IF(AND(H136=0,H137=0,H138=0),\"\",\" almost\")&\" at the steady-state.\"","v":"are almost at the steady-state."}]
[null,null,null,null,2,{"$f":"G104","v":0.46210833876377388},{"$f":"E132","v":0.46210766844653867},{"$f":"F138-G138","v":6.7031723521848718E-7}]
[]
[null,null,null,null,"Expected M for the next flip in the steady-state"]
[null,null,null,null,{"$f":"E130*(B5-C5)+E131*(C6-A6)+E132*(A7-B7)","v":-8.6952866935818274E-3}]

{"sheet":"MCParr","index":5,"n_rows":141,"n_cols":19,"rows_kept":141,"cut":false,"hidden":true,"merged":[[2,4,2,6]]}
["Analytical Solution via Markov Chain"]
[null,null,null,null,null,null,null,null,"Initial M",0]
["epsilon",5.0000000000000001E-3,null,null,"Mod(M,3)"]
["Transition Matrix: Parrondo",null,null,null,0,1,2,null,"Flip #","Expected M"]
[0,{"$f":"0.5*(0.5-$B$3)+0.5*(0.1-$B$3)","v":0.29499999999999998},{"$f":"0.5*(0.5+$B$3)+0.5*(0.9+$B$3)","v":0.70500000000000007},null,{"$f":"IF(MOD($J$2,3)=0,1,0)","v":1},{"$f":"IF(MOD($J$2,3)=1,1,0)","v":0},{"$f":"IF(MOD($J$2,3)=2,1,0)","v":0},null,1,{"$f":"E5*($B$5-$C$5)+F5*($C$6-$A$6)+G5*($A$7-$B$7)+J2","v":-0.41000000000000009}]
[{"$f":"0.5*(0.5+$B$3)+0.5*(0.25+$B$3)","v":0.38},0,{"$f":"0.5*(0.5-$B$3)+0.5*(0.75-$B$3)","v":0.62},null,{"$f":"MMULT(E5:G5,$A$5:$C$7)","v":0},0.29499999999999998,0.70500000000000007,null,2,{"$f":"E6*($B$5-$C$5)+F6*($C$6-$A$6)+G6*($A$7-$B$7)+J5","v":-0.1700000000000001}]
[{"$f":"0.5*(0.5-$B$3)+0.5*(0.75-$B$3)","v":0.62},{"$f":"0.5*(0.5+$B$3)+0.5*(0.25+$B$3)","v":0.38},0,null,{"$f":"MMULT(E6:G6,$A$5:$C$7)","v":0.54920000000000002},0.26790000000000003,0.18289999999999998,null,3,-0.28698000000000018]
[null,null,null,null,{"$f":"MMULT(E7:G7,$A$5:$C$7)","v":0.2152},0.231516,0.55328400000000011,null,4,-0.18686000000000019]
[null,null,null,null,{"$f":"MMULT(E8:G8,$A$5:$C$7)","v":0.43101216000000009},0.27373192000000002,0.29525592000000001,null,5,-0.22701790400000027]
["Transition Matrix: Game A",null,null,null,{"$f":"MMULT(E9:G9,$A$5:$C$7)","v":0.28707680000000002},0.23934583680000004,0.4735773632000001,null,6,-0.17361782400000028]
[null,null,null,null,{"$f":"MMULT(E10:G10,$A$5:$C$7)","v":0.3845693831680001},0.26464705401600003,0.35078356281600009,null,7,-0.18358792305920033]
[0,{"$f":"0.5-$B$3","v":0.495},{"$f":"0.5+$B$3","v":0.505},null,{"$f":"MMULT(E11:G11,$A$5:$C$7)","v":0.31805168947200008},0.24674572190464006,0.43520258862336014,null,8,-0.15032152121600034]
[{"$f":"0.5+$B$3","v":0.505},0,{"$f":"0.5-$B$3","v":0.495},null,{"$f":"MMULT(E12:G12,$A$5:$C$7)","v":0.36358897927024647},0.25920223207111687,0.37720878865863694,null,9,-0.1466543577416605]
[{"$f":"0.5-$B$3","v":0.495},{"$f":"0.5+$B$3","v":0.505},0,null,{"$f":"MMULT(E13:G13,$A$5:$C$7)","v":0.3323662971553793},0.25059808857500476,0.41703561426961622,null,10,-0.122692450892657]
[null,null,null,null,{"$f":"MMULT(E14:G14,$A$5:$C$7)","v":0.35378935450566384},0.25652159108329103,0.3896890544110454,null,11,-0.11265553132133846]
["Transition Matrix: Game B",null,null,null,{"$f":"MMULT(E15:G15,$A$5:$C$7)","v":0.33908541834649875},0.25244970025536806,0.40846488139813347,null,12,-9.3061053246562594E-2]
[null,null,null,null,{"$f":"MMULT(E16:G16,$A$5:$C$7)","v":0.34917911256388262},0.25524685334350783,0.39557403409260983,null,13,-8.002747641308626E-2]
[0,{"$f":"0.1-$B$3","v":9.5000000000000001E-2},{"$f":"0.9+$B$3","v":0.90500000000000003},null,{"$f":"MMULT(E17:G17,$A$5:$C$7)","v":0.34224970540795108},0.2533259711615371,0.40442432343051216,null,14,-6.2489784928254424E-2]
[{"$f":"0.25+$B$3","v":0.255},0,{"$f":"0.75-$B$3","v":0.745},null,{"$f":"MMULT(E18:G18,$A$5:$C$7)","v":0.34700694956830158},0.25464490599894019,0.39834814443275857,null,15,-4.8044302147650417E-2]
[{"$f":"0.75-$B$3","v":0.745},{"$f":"0.25+$B$3","v":0.255},0,null,{"$f":"MMULT(E19:G19,$A$5:$C$7)","v":0.34374091382790761},0.25373934500709722,0.40251974116499556,null,16,-3.1475896135790299E-2]
[null,null,null,null,{"$f":"MMULT(E20:G20,$A$5:$C$7)","v":0.34598319062499416},0.25436107122193108,0.3996557381530752,null,17,-1.6364970042036422E-2]
[null,null,null,null,{"$f":"MMULT(E21:G21,$A$5:$C$7)","v":0.34444376471924043},0.25393422173254188,0.40162201354821814,null,18,-2.5341710954261687E-4]
[null,null,null,null,{"$f":"MMULT(E22:G22,$A$5:$C$7)","v":0.34550065265826113},0.2542272757404988,0.40027207160124051,null,19,1.5171158662587725E-2]
[null,null,null,null,{"$f":"MMULT(E23:G23,$A$5:$C$7)","v":0.34477504917415869},0.25402607974265839,0.40119887108318342,null,20,3.1067376699384655E-2]
[null,null,null,null,{"$f":"MMULT(E24:G24,$A$5:$C$7)","v":0.34527321037378389},0.25416421051798649,0.40056257910823012,null,21,4.6639789956425204E-2]
[null,null,null,null,{"$f":"MMULT(E25:G25,$A$5:$C$7)","v":0.34493119904393754},0.25406937712139371,0.40099942383466924,null,22,6.2434510577865879E-2]
[null,null,null,null,{"$f":"MMULT(E26:G26,$A$5:$C$7)","v":0.34516600608362458},0.25413448477513589,0.40069950914124008,null,23,7.8076606623509986E-2]
[null,null,null,null,{"$f":"MMULT(E27:G27,$A$5:$C$7)","v":0.34500479988212052},0.25408978526834047,0.40090541484953957,null,24,9.3823486700131742E-2]
[null,null,null,null,{"$f":"MMULT(E28:G28,$A$5:$C$7)","v":0.34511547560868389},0.25412047360805057,0.4007640507832661,null,25,0.10949842755448731]
[null,null,null,null,{"$f":"MMULT(E29:G29,$A$5:$C$7)","v":0.34503949145668422},0.25409940460220287,0.40086110394111352,null,26,0.12522275810764266]
[null,null,null,null,{"$f":"MMULT(E30:G30,$A$5:$C$7)","v":0.34509165819232746},0.25411386947734499,0.40079447233032817,null,27,0.14091318028262992]
[null,null,null,null,{"$f":"MMULT(E31:G31,$A$5:$C$7)","v":0.34505584324619454},0.25410393865226127,0.4008402181015448,null,28,0.15662688217260357]
[null,null,null,null,{"$f":"MMULT(E32:G32,$A$5:$C$7)","v":0.34508043191081705},0.25411075663621441,0.40080881145296915,null,29,0.1723246014305726]
[null,null,null,null,{"$f":"MMULT(E33:G33,$A$5:$C$7)","v":0.34506355062260236},0.25410607576581934,0.40083037361157897,null,30,0.18803329352588116]
[null,null,null,null,{"$f":"MMULT(E34:G34,$A$5:$C$7)","v":0.34507514043019027},0.2541092894060677,0.4008155701637427,null,31,0.20373445224625761]
[null,null,null,null,{"$f":"MMULT(E35:G35,$A$5:$C$7)","v":0.34506718347582621},0.25410708308912838,0.40082573343504613,null,32,0.21944078298697073]
[null,null,null,null,{"$f":"MMULT(E36:G36,$A$5:$C$7)","v":0.34507264630359735},0.25410859783068629,0.40081875586571714,null,33,0.23514356288963262]
[null,null,null,null,{"$f":"MMULT(E37:G37,$A$5:$C$7)","v":0.34506889581240541},0.25410755788853373,0.40082354629906164,null,34,{"$f":"E38*($B$5-$C$5)+F38*($C$6-$A$6)+G38*($A$7-$B$7)+J37","v":0.25084878061156923}]
[null,null,null,null,{"$f":"MMULT(E38:G38,$A$5:$C$7)","v":0.34507147070306105},0.25410827185830298,0.40082025743863675,null,35,0.26655232465457968]
[null,null,null,null,{"$f":"MMULT(E39:G39,$A$5:$C$7)","v":0.3450697029181099},0.25410778168408499,0.40082251539780589,null,36,0.28225701775780843]
[null,null,null,null,{"$f":"MMULT(E40:G40,$A$5:$C$7)","v":0.34507091658659195},0.25410811821200863,0.4008209652014002,null,37,0.2979609219765238]
[null,null,null,null,{"$f":"MMULT(E41:G41,$A$5:$C$7)","v":0.34507008334543143},0.25410788716957672,0.40082202948499268,null,38,0.3136653678019935]
[null,null,null,null,{"$f":"MMULT(E42:G42,$A$5:$C$7)","v":0.3450706554051346},0.2541080457911995,0.40082129880366674,null,39,0.32936944178865618]
[null,null,null,null,{"$f":"MMULT(E43:G43,$A$5:$C$7)","v":0.34507026265892921},0.25410793688990807,0.40082180045116356,null,40,0.34507377106035236]
[null,null,null,null,{"$f":"MMULT(E44:G44,$A$5:$C$7)","v":0.34507053229788648},0.25410801165582625,0.4008214560462881,null,41,0.36077792506672635]
[null,null,null,null,{"$f":"MMULT(E45:G45,$A$5:$C$7)","v":0.34507034717791263},0.25410796032546601,0.40082169249662225,null,42,0.3764821994010833]
[null,null,null,null,{"$f":"MMULT(E46:G46,$A$5:$C$7)","v":0.34507047427158288},0.25410799556620067,0.40082153016221733,null,43,0.39218639112455461]
[null,null,null,null,{"$f":"MMULT(E47:G47,$A$5:$C$7)","v":0.34507038701573101},0.25410797137175956,0.40082164161251038,null,44,0.40789063956432964]
[null,null,null,null,{"$f":"MMULT(E48:G48,$A$5:$C$7)","v":0.34507044692102506},0.25410798798239459,0.40082156509658129,null,45,0.42359484906566353]
[null,null,null,null,{"$f":"MMULT(E49:G49,$A$5:$C$7)","v":0.34507040579319037},0.25410797657840328,0.40082161762840735,null,46,0.43929908530008999]
[null,null,null,null,{"$f":"MMULT(E50:G50,$A$5:$C$7)","v":0.34507043402940579},0.25410798440778593,0.40082158156280928,null,47,0.45500330318097643]
[null,null,null,null,{"$f":"MMULT(E51:G51,$A$5:$C$7)","v":0.34507041464390042},0.25410797903254223,0.4008216063235584,null,48,0.47070753366244139]
[null,null,null,null,{"$f":"MMULT(E52:G52,$A$5:$C$7)","v":0.34507042795297227},0.2541079827229028,0.40082158932412604,null,49,0.48641175549300963]
[null,null,null,null,{"$f":"MMULT(E53:G53,$A$5:$C$7)","v":0.34507041881566125},0.2541079801892947,0.40082160099504521,null,50,0.50211598326283013]
[null,null,null,null,{"$f":"MMULT(E54:G54,$A$5:$C$7)","v":0.34507042508885999},0.25410798192873724,0.40082159298240394,null,51,0.51782020695507136]
[null,null,null,null,{"$f":"MMULT(E55:G55,$A$5:$C$7)","v":0.34507042078201061},0.25410798073452717,0.40082159848346344,null,52,0.5335244334467647]
[null,null,null,null,{"$f":"MMULT(E56:G56,$A$5:$C$7)","v":0.34507042373886765},0.25410798155440922,0.40082159470672435,null,53,0.549228658016501]
[null,null,null,null,{"$f":"MMULT(E57:G57,$A$5:$C$7)","v":0.34507042170884461},0.25410798099152121,0.4008215972996354,null,54,0.56493288390575225]
[null,null,null,null,{"$f":"MMULT(E58:G58,$A$5:$C$7)","v":0.34507042310255204},0.25410798137797058,0.4008215955194786,null,55,0.5806371088890937]
[null,null,null,null,{"$f":"MMULT(E59:G59,$A$5:$C$7)","v":0.34507042214570555},0.2541079811126547,0.40082159674164097,null,56,0.59634133449438531]
[null,null,null,null,{"$f":"MMULT(E60:G60,$A$5:$C$7)","v":0.34507042280262618},0.25410798129480672,0.40082159590256838,null,57,0.6120455596726786]
[null,null,null,null,{"$f":"MMULT(E61:G61,$A$5:$C$7)","v":0.34507042235161894},0.2541079811697507,0.40082159647863164,null,58,0.6277497851441266]
[null,null,null,null,{"$f":"MMULT(E62:G62,$A$5:$C$7)","v":0.34507042266125687},0.25410798125560763,0.40082159608313683,null,59,0.64345401041430994]
[null,null,null,null,{"$f":"MMULT(E63:G63,$A$5:$C$7)","v":0.34507042244867575},0.25410798119666278,0.40082159635466286,null,60,0.65915823582267097]
[null,null,null,null,{"$f":"MMULT(E64:G64,$A$5:$C$7)","v":0.34507042259462284},0.25410798123713119,0.40082159616824736,null,61,0.67486246113616644]
[null,null,null,null,{"$f":"MMULT(E65:G65,$A$5:$C$7)","v":0.34507042249442321},0.25410798120934774,0.40082159629623049,null,62,0.69056668651479169]
[null,null,null,null,{"$f":"MMULT(E66:G66,$A$5:$C$7)","v":0.34507042256321507},0.25410798122842243,0.40082159620836399,null,63,0.70627091184870228]
[null,null,null,null,{"$f":"MMULT(E67:G67,$A$5:$C$7)","v":0.34507042251598619},0.25410798121532674,0.40082159626868857,null,64,0.72197513721331164]
[null,null,null,null,{"$f":"MMULT(E68:G68,$A$5:$C$7)","v":0.34507042254841108},0.25410798122431755,0.40082159622727287,null,65,0.73767936255684474]
[null,null,null,null,{"$f":"MMULT(E69:G69,$A$5:$C$7)","v":0.34507042252614983},0.25410798121814493,0.40082159625570668,null,66,{"$f":"E70*($B$5-$C$5)+F70*($C$6-$A$6)+G70*($A$7-$B$7)+J69","v":0.75338358791484761}]
[null,null,null,null,{"$f":"MMULT(E70:G70,$A$5:$C$7)","v":0.34507042254143322},0.25410798122238276,0.40082159623618552,null,67,0.76908781326291631]
[null,null,null,null,{"$f":"MMULT(E71:G71,$A$5:$C$7)","v":0.34507042253094045},0.25410798121947331,0.40082159624958774,null,68,0.78479203861780533]
[null,null,null,null,{"$f":"MMULT(E72:G72,$A$5:$C$7)","v":0.34507042253814424},0.25410798122147077,0.40082159624038649,null,69,0.80049626396801188]
[null,null,null,null,{"$f":"MMULT(E73:G73,$A$5:$C$7)","v":0.34507042253319853},0.25410798122009942,0.40082159624670355,null,70,0.81620048932143319]
[null,null,null,null,{"$f":"MMULT(E74:G74,$A$5:$C$7)","v":0.34507042253659398},0.25410798122104095,0.40082159624236663,null,71,0.83190471467264748]
[null,null,null,null,{"$f":"MMULT(E75:G75,$A$5:$C$7)","v":0.3450704225342629},0.25410798122039452,0.40082159624534419,null,72,0.84760894002537701]
[null,null,null,null,{"$f":"MMULT(E76:G76,$A$5:$C$7)","v":0.34507042253586329},0.25410798122083833,0.40082159624329994,null,73,0.86331316537706626]
[null,null,null,null,{"$f":"MMULT(E77:G77,$A$5:$C$7)","v":0.3450704225347645},0.25410798122053363,0.40082159624470337,null,74,0.87901739072946961]
[null,null,null,null,{"$f":"MMULT(E78:G78,$A$5:$C$7)","v":0.34507042253551889},0.2541079812207428,0.40082159624373986,null,75,0.89472161608138268]
[null,null,null,null,{"$f":"MMULT(E79:G79,$A$5:$C$7)","v":0.34507042253500098},0.25410798122059919,0.40082159624440139,null,76,0.91042584143363237]
[null,null,null,null,{"$f":"MMULT(E80:G80,$A$5:$C$7)","v":0.34507042253535658},0.25410798122069783,0.4008215962439472,null,77,0.92613006678565091]
[null,null,null,null,{"$f":"MMULT(E81:G81,$A$5:$C$7)","v":0.34507042253511244},0.25410798122063016,0.40082159624425906,null,78,0.94183429213782821]
[null,null,null,null,{"$f":"MMULT(E82:G82,$A$5:$C$7)","v":0.34507042253528009},0.25410798122067657,0.40082159624404501,null,79,0.95753851748989649]
[null,null,null,null,{"$f":"MMULT(E83:G83,$A$5:$C$7)","v":0.34507042253516501},0.25410798122064471,0.400821596244192,null,80,0.9732427428420396]
[null,null,null,null,{"$f":"MMULT(E84:G84,$A$5:$C$7)","v":0.345070422535244},0.25410798122066663,0.40082159624409108,null,81,0.98894696819413141]
[null,null,null,null,{"$f":"MMULT(E85:G85,$A$5:$C$7)","v":0.34507042253518977},0.25410798122065159,0.40082159624416036,null,82,1.0046511935462585]
[null,null,null,null,{"$f":"MMULT(E86:G86,$A$5:$C$7)","v":0.34507042253522702},0.25410798122066192,0.40082159624411284,null,83,1.0203554188983615]
[null,null,null,null,{"$f":"MMULT(E87:G87,$A$5:$C$7)","v":0.34507042253520148},0.25410798122065481,0.40082159624414548,null,84,1.0360596442504808]
[null,null,null,null,{"$f":"MMULT(E88:G88,$A$5:$C$7)","v":0.34507042253521902},0.2541079812206597,0.40082159624412306,null,85,1.0517638696025888]
[null,null,null,null,{"$f":"MMULT(E89:G89,$A$5:$C$7)","v":0.34507042253520698},0.25410798122065636,0.40082159624413849,null,86,1.0674680949547046]
[null,null,null,null,{"$f":"MMULT(E90:G90,$A$5:$C$7)","v":0.34507042253521525},0.2541079812206587,0.40082159624412789,null,87,1.0831723203068151]
[null,null,null,null,{"$f":"MMULT(E91:G91,$A$5:$C$7)","v":0.34507042253520959},0.25410798122065709,0.40082159624413516,null,88,1.0988765456589293]
[null,null,null,null,{"$f":"MMULT(E92:G92,$A$5:$C$7)","v":0.34507042253521347},0.2541079812206582,0.40082159624413016,null,89,1.1145807710110409]
[null,null,null,null,{"$f":"MMULT(E93:G93,$A$5:$C$7)","v":0.34507042253521081},0.25410798122065748,0.4008215962441336,null,90,1.1302849963631543]
[null,null,null,null,{"$f":"MMULT(E94:G94,$A$5:$C$7)","v":0.3450704225352127},0.25410798122065797,0.40082159624413127,null,91,1.1459892217152665]
[null,null,null,null,{"$f":"MMULT(E95:G95,$A$5:$C$7)","v":0.34507042253521142},0.25410798122065764,0.40082159624413294,null,92,1.1616934470673796]
[null,null,null,null,{"$f":"MMULT(E96:G96,$A$5:$C$7)","v":0.34507042253521231},0.25410798122065792,0.40082159624413183,null,93,1.1773976724194921]
[null,null,null,null,{"$f":"MMULT(E97:G97,$A$5:$C$7)","v":0.34507042253521175},0.2541079812206577,0.40082159624413261,null,94,1.193101897771605]
[null,null,null,null,{"$f":"MMULT(E98:G98,$A$5:$C$7)","v":0.34507042253521214},0.25410798122065786,0.40082159624413205,null,95,1.2088061231237175]
[null,null,null,null,{"$f":"MMULT(E99:G99,$A$5:$C$7)","v":0.34507042253521186},0.25410798122065775,0.40082159624413249,null,96,1.2245103484758302]
[null,null,null,null,{"$f":"MMULT(E100:G100,$A$5:$C$7)","v":0.34507042253521208},0.25410798122065781,0.40082159624413216,null,97,1.2402145738279429]
[null,null,null,null,{"$f":"MMULT(E101:G101,$A$5:$C$7)","v":0.34507042253521192},0.25410798122065781,0.40082159624413238,null,98,1.2559187991800556]
[null,null,null,null,{"$f":"MMULT(E102:G102,$A$5:$C$7)","v":0.34507042253521203},0.25410798122065781,0.40082159624413227,null,99,1.2716230245321682]
[null,null,null,null,{"$f":"MMULT(E103:G103,$A$5:$C$7)","v":0.34507042253521197},0.25410798122065781,0.40082159624413238,null,100,1.2873272498842809,"Final M"]
[]
[]
[null,null,null,null,"Finding Exact Steady-state probabilities (as # flips --> infinity)",null,null,null,null,null,null,null,"Table 1 in paper:"]
[null,null,null,null,"Transpose the transition matrix:",null,null,null,null,null,null,null,"Epsilon = 0.005",null,null,"Steady-state probabilities",null,null,"Next Flip"]
[null,null,null,null,{"$f":"TRANSPOSE(A5:C7)","v":0},0.38,0.62,null,null,null,null,null,"Game A",null,null,0,1,2,"Expected M"]
[null,null,null,null,0.29499999999999998,0,0.38,null,null,null,null,null,0,0.495,0.505,0.33333333333333331,0.33333333333333331,0.33333333333333331,{"$f":"P110*(N110-O110)+Q110*(O111-M111)+R110*(M112-N112)","v":-1.0000000000000009E-2}]
[null,null,null,null,0.70500000000000007,0.62,0,null,null,null,null,null,0.505,0,0.495,0.33333333333333331,0.33333333333333331,0.33333333333333331]
[null,null,null,null,null,null,null,null,null,null,null,null,0.495,0.505,0,"lose 0.01","lose 0.01","lose 0.01"]
[null,null,null,null,"Subtract the identity matrix:"]
[null,null,null,null,-1,{"$f":"F109","v":0.38},{"$f":"G109","v":0.62},null,null,null,null,null,"Game B"]
[null,null,null,null,{"$f":"E110","v":0.29499999999999998},-1,{"$f":"G110","v":0.38},null,null,null,null,null,0,9.5000000000000001E-2,0.90500000000000003,0.38461538461538458,0.15384615384615383,0.46153846153846156,{"$f":"P115*(N115-O115)+Q115*(O116-M116)+R115*(M117-N117)","v":-1.0000000000000009E-2}]
[null,null,null,null,{"$f":"E111","v":0.70500000000000007},{"$f":"F111","v":0.62},-1,null,null,null,null,null,0.255,0,0.745,{"$f":"P115","v":0.38461538461538458},{"$f":"Q115","v":0.15384615384615383},{"$f":"R115","v":0.46153846153846156}]
[null,null,null,null,null,null,null,null,null,null,null,null,0.745,0.255,0,"lose 0.81","win 0.49","win 0.49"]
[null,null,null,null,"Replace the last row (any row would work) with the "]
[null,null,null,null,"constraint that the probabilities sum to one:",null,null,null,null,null,null,null,"Randomly Mixed"]
[null,null,null,null,{"$f":"E114","v":-1},0.38,0.62,"p0",null,0,null,null,0,0.29499999999999998,0.70500000000000007,0.34507042253521197,0.25410798122065781,0.40082159624413238,{"$f":"P120*(N120-O120)+Q120*(O121-M121)+R120*(M122-N122)","v":1.5704225352112688E-2}]
[null,null,null,null,0.29499999999999998,-1,0.38,"p1","=",0,null,null,0.38,0,0.62,{"$f":"P120","v":0.34507042253521197},{"$f":"Q120","v":0.25410798122065781},{"$f":"R120","v":0.40082159624413238}]
[null,null,null,null,1,1,1,"p2",null,1,null,null,0.62,0.38,0,"lose 0.41","win 0.24","win 0.24"]
[]
[null,null,null,null,"Invert the matrix:"]
[null,null,null,null,{"$f":"MINVERSE(E120:G122)","v":-0.62296858071505956},0.10834236186348861,0.34507042253521125]
[null,null,null,null,3.8371253159985574E-2,-0.73131094257854823,0.25410798122065725]
[null,null,null,null,0.58459732755507399,0.62296858071505956,0.40082159624413144]
[]
[null,null,null,null,"Multiply both sides by the inverse to find the steady-state solution:"]
[null,null,null,null,{"$f":"MMULT(E125:G127,J120:J122)","v":0.34507042253521125}]
[null,null,null,null,0.25410798122065725]
[null,null,null,null,0.40082159624413144]
[]
[null,null,null,null,"Difference between probabilties at 100th flip and steady-state:"]
[null,null,null,null,"Position","100th flip","Steady-state","Difference","Conclusion"]
[null,null,null,null,0,{"$f":"E104","v":0.34507042253521197},{"$f":"E130","v":0.34507042253521125},{"$f":"F136-G136","v":7.2164496600635175E-16},"After 100 flips, the probabilities"]
[null,null,null,null,1,{"$f":"F104","v":0.25410798122065781},{"$f":"E131","v":0.25410798122065725},{"$f":"F137-G137","v":5.5511151231257827E-16},{"$f":"\"are\"&IF(AND(H136=0,H137=0,H138=0),\"\",\" almost\")&\" at the steady-state.\"","v":"are almost at the steady-state."}]
[null,null,null,null,2,{"$f":"G104","v":0.40082159624413238},{"$f":"E132","v":0.40082159624413144},{"$f":"F138-G138","v":9.4368957093138306E-16}]
[]
[null,null,null,null,"Expected M for the next flip in the steady-state"]
[null,null,null,null,{"$f":"E130*(B5-C5)+E131*(C6-A6)+E132*(A7-B7)","v":1.5704225352112633E-2}]

{"sheet":"Optimal","index":6,"n_rows":107,"n_cols":27,"rows_kept":107,"cut":false,"hidden":true,"merged":[]}
["Understanding Parrondo's Paradox",null,null,null,null,null,null,null,null,null,null,null,null,"Playing OPTIMALLY",null,null,null,null,null,null,null,null,null,null,null,1,"Random"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Play sequence is AB then ABBABBABB repeatedly",null,null,null,null,null,null,null,null,null,null,null,null,"ABAB"]
["Game A",null,null,null,null,null,"Game B",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"AABB"]
["epsilon",5.0000000000000001E-3,"amount of bias in the coin",null,null,null,"Coin 1",null,null,"Coin 2"]
["Prob Heads",{"$f":"0.5-B4","v":0.495},null,null,null,null,"Prob Heads",{"$f":"0.1-B4","v":9.5000000000000001E-2},null,"Prob Heads",{"$f":"0.75-B4","v":0.745}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"IF Game B is chosen",null,null,"Game B","Actual"]
["Flip #","Result","End M",null,null,null,"Flip #","Start M","M/3 Remainder","Coin","Result","End M",null,"Flip #","Game","Start M","M/3 Remainder","Coin","Result","Result","End M"]
[1,{"$f":"IF(RAND()<=$B$5,1,0)","v":1},{"$f":"IF(B8=1,1,-1)","v":1},null,null,null,1,0,{"$f":"MOD(H8,3)","v":0},{"$f":"IF(I8=0,1,2)","v":1},{"$f":"IF(I8=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":1},{"$f":"IF(K8=1,1,-1)","v":1},null,1,"A",0,{"$f":"MOD(P8,3)","v":0},1,{"$f":"IF(RAND()<=$H$5,1,0)","v":0},{"$f":"IF(O8=\"A\",B8,S8)","v":1},{"$f":"IF(T8=1,1,-1)","v":1}]
[2,0,{"$f":"IF(B9=1,1+C8,-1+C8)","v":0},null,null,null,2,{"$f":"L8","v":1},1,2,1,{"$f":"IF(K9=1,1+L8,-1+L8)","v":2},null,2,"B",{"$f":"U8","v":1},1,{"$f":"IF(Q9=0,1,2)","v":2},{"$f":"IF(Q9=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":1},1,{"$f":"IF(T9=1,1+U8,-1+U8)","v":2}]
[3,0,-1,null,null,null,3,2,2,2,1,3,null,3,"A",2,2,2,0,0,1]
[4,1,0,null,null,null,4,3,0,1,0,2,null,4,"B",1,1,2,1,1,2]
[5,0,-1,null,null,null,5,2,2,2,1,3,null,5,"B",2,2,2,1,1,3]
[6,0,-2,null,null,null,6,3,0,1,0,2,null,6,"A",3,0,1,0,0,2]
[7,0,-3,null,null,null,7,2,2,2,1,3,null,7,"B",2,2,2,0,0,1]
[8,1,-2,null,null,null,8,3,0,1,1,4,null,8,"B",1,1,2,0,0,0]
[9,0,-3,null,null,null,9,4,1,2,1,5,null,9,"A",0,0,1,0,0,-1]
[10,1,-2,null,null,null,10,5,2,2,0,4,null,10,"B",-1,2,2,1,1,0]
[11,0,-3,null,null,null,11,4,1,2,1,5,null,11,"B",0,0,1,0,0,-1]
[12,0,-4,null,null,null,12,5,2,2,0,4,null,12,"A",-1,2,2,1,0,-2]
[13,0,-5,null,null,null,13,4,1,2,1,5,null,13,"B",-2,1,2,0,0,-3]
[14,1,-4,null,null,null,14,5,2,2,1,6,null,14,"B",-3,0,1,0,0,-4]
[15,0,-5,null,null,null,15,6,0,1,0,5,null,15,"A",-4,2,2,0,0,-5]
[16,1,-4,null,null,null,16,5,2,2,1,6,null,16,"B",-5,1,2,0,0,-6]
[17,1,-3,null,null,null,17,6,0,1,0,5,null,17,"B",-6,0,1,0,0,-7]
[18,1,-2,null,null,null,18,5,2,2,1,6,null,18,"A",-7,2,2,1,1,-6]
[19,1,-1,null,null,null,19,6,0,1,0,5,null,19,"B",-6,0,1,0,0,-7]
[20,0,-2,null,null,null,20,5,2,2,1,6,null,20,"B",-7,2,2,1,1,-6]
[21,1,-1,null,null,null,21,6,0,1,0,5,null,21,"A",-6,0,1,0,1,-5]
[22,0,-2,null,null,null,22,5,2,2,1,6,null,22,"B",-5,1,2,1,1,-4]
[23,0,-3,null,null,null,23,6,0,1,0,5,null,23,"B",-4,2,2,0,0,-5]
[24,0,-4,null,null,null,24,5,2,2,1,6,null,24,"A",-5,1,2,1,0,-6]
[25,0,-5,null,null,null,25,6,0,1,0,5,null,25,"B",-6,0,1,0,0,-7]
[26,1,-4,null,null,null,26,5,2,2,1,6,null,26,"B",-7,2,2,1,1,-6]
[27,0,-5,null,null,null,27,6,0,1,0,5,null,27,"A",-6,0,1,0,0,-7]
[28,1,-4,null,null,null,28,5,2,2,1,6,null,28,"B",-7,2,2,1,1,-6]
[29,1,-3,null,null,null,29,6,0,1,0,5,null,29,"B",-6,0,1,1,1,-5]
[30,0,-4,null,null,null,30,5,2,2,1,6,null,30,"A",-5,1,2,1,0,-6]
[31,0,-5,null,null,null,31,6,0,1,1,7,null,31,"B",-6,0,1,0,0,-7]
[32,0,-6,null,null,null,32,7,1,2,0,6,null,32,"B",-7,2,2,1,1,-6]
[33,0,-7,null,null,null,33,6,0,1,0,5,null,33,"A",-6,0,1,0,0,-7]
[34,0,-8,null,null,null,34,5,2,2,0,4,null,34,"B",-7,2,2,1,1,-6]
[35,1,-7,null,null,null,35,4,1,2,1,5,null,35,"B",-6,0,1,0,0,-7]
[36,0,-8,null,null,null,36,5,2,2,0,4,null,36,"A",-7,2,2,1,0,-8]
[37,0,-9,null,null,null,37,4,1,2,1,5,null,37,"B",-8,1,2,1,1,-7]
[38,1,-8,null,null,null,38,5,2,2,1,6,null,38,"B",-7,2,2,1,1,-6]
[39,0,-9,null,null,null,39,6,0,1,1,7,null,39,"A",-6,0,1,0,0,-7]
[40,1,-8,null,null,null,40,7,1,2,1,8,null,40,"B",-7,2,2,1,1,-6]
[41,1,-7,null,null,null,41,8,2,2,1,9,null,41,"B",-6,0,1,0,0,-7]
[42,0,-8,null,null,null,42,9,0,1,0,8,null,42,"A",-7,2,2,1,0,-8]
[43,1,-7,null,null,null,43,8,2,2,1,9,null,43,"B",-8,1,2,1,1,-7]
[44,1,-6,null,null,null,44,9,0,1,0,8,null,44,"B",-7,2,2,1,1,-6]
[45,1,-5,null,null,null,45,8,2,2,1,9,null,45,"A",-6,0,1,0,1,-5]
[46,0,-6,null,null,null,46,9,0,1,0,8,null,46,"B",-5,1,2,0,0,-6]
[47,1,-5,null,null,null,47,8,2,2,1,9,null,47,"B",-6,0,1,0,0,-7]
[48,1,-4,null,null,null,48,9,0,1,0,8,null,48,"A",-7,2,2,0,1,-6]
[49,1,-3,null,null,null,49,8,2,2,0,7,null,49,"B",-6,0,1,0,0,-7]
[50,1,-2,null,null,null,50,7,1,2,1,8,null,50,"B",-7,2,2,1,1,-6]
[51,1,-1,null,null,null,51,8,2,2,1,9,null,51,"A",-6,0,1,0,1,-5]
[52,1,0,null,null,null,52,9,0,1,0,8,null,52,"B",-5,1,2,0,0,-6]
[53,1,1,null,null,null,53,8,2,2,0,7,null,53,"B",-6,0,1,0,0,-7]
[54,1,2,null,null,null,54,7,1,2,1,8,null,54,"A",-7,2,2,1,1,-6]
[55,0,1,null,null,null,55,8,2,2,1,9,null,55,"B",-6,0,1,0,0,-7]
[56,1,2,null,null,null,56,9,0,1,0,8,null,56,"B",-7,2,2,1,1,-6]
[57,1,3,null,null,null,57,8,2,2,1,9,null,57,"A",-6,0,1,0,1,-5]
[58,0,2,null,null,null,58,9,0,1,0,8,null,58,"B",-5,1,2,0,0,-6]
[59,1,3,null,null,null,59,8,2,2,0,7,null,59,"B",-6,0,1,0,0,-7]
[60,1,4,null,null,null,60,7,1,2,0,6,null,60,"A",-7,2,2,1,1,-6]
[61,1,5,null,null,null,61,6,0,1,0,5,null,61,"B",-6,0,1,0,0,-7]
[62,1,6,null,null,null,62,5,2,2,1,6,null,62,"B",-7,2,2,1,1,-6]
[63,1,7,null,null,null,63,6,0,1,1,7,null,63,"A",-6,0,1,0,1,-5]
[64,0,6,null,null,null,64,7,1,2,0,6,null,64,"B",-5,1,2,1,1,-4]
[65,{"$f":"IF(RAND()<=$B$5,1,0)","v":1},7,null,null,null,65,6,{"$f":"MOD(H72,3)","v":0},{"$f":"IF(I72=0,1,2)","v":1},{"$f":"IF(I72=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":1},7,null,65,"B",-4,{"$f":"MOD(P72,3)","v":2},2,0,{"$f":"IF(O72=\"A\",B72,S72)","v":0},-5]
[66,0,{"$f":"IF(B73=1,1+C72,-1+C72)","v":6},null,null,null,66,{"$f":"L72","v":7},1,2,1,{"$f":"IF(K73=1,1+L72,-1+L72)","v":8},null,66,"A",{"$f":"U72","v":-5},1,{"$f":"IF(Q73=0,1,2)","v":2},{"$f":"IF(Q73=0,IF(RAND()<=$H$5,1,0),IF(RAND()<=$K$5,1,0))","v":0},0,{"$f":"IF(T73=1,1+U72,-1+U72)","v":-6}]
[67,1,7,null,null,null,67,8,2,2,1,9,null,67,"B",-6,0,1,1,1,-5]
[68,0,6,null,null,null,68,9,0,1,0,8,null,68,"B",-5,1,2,1,1,-4]
[69,0,5,null,null,null,69,8,2,2,0,7,null,69,"A",-4,2,2,1,0,-5]
[70,0,4,null,null,null,70,7,1,2,1,8,null,70,"B",-5,1,2,1,1,-4]
[71,0,3,null,null,null,71,8,2,2,0,7,null,71,"B",-4,2,2,1,1,-3]
[72,1,4,null,null,null,72,7,1,2,1,8,null,72,"A",-3,0,1,0,1,-2]
[73,0,3,null,null,null,73,8,2,2,0,7,null,73,"B",-2,1,2,1,1,-1]
[74,0,2,null,null,null,74,7,1,2,1,8,null,74,"B",-1,2,2,0,0,-2]
[75,1,3,null,null,null,75,8,2,2,1,9,null,75,"A",-2,1,2,1,1,-1]
[76,1,4,null,null,null,76,9,0,1,1,10,null,76,"B",-1,2,2,1,1,0]
[77,0,3,null,null,null,77,10,1,2,1,11,null,77,"B",0,0,1,0,0,-1]
[78,1,4,null,null,null,78,11,2,2,1,12,null,78,"A",-1,2,2,1,1,0]
[79,1,5,null,null,null,79,12,0,1,0,11,null,79,"B",0,0,1,0,0,-1]
[80,0,4,null,null,null,80,11,2,2,1,12,null,80,"B",-1,2,2,1,1,0]
[81,0,3,null,null,null,81,12,0,1,0,11,null,81,"A",0,0,1,0,0,-1]
[82,0,2,null,null,null,82,11,2,2,1,12,null,82,"B",-1,2,2,1,1,0]
[83,1,3,null,null,null,83,12,0,1,0,11,null,83,"B",0,0,1,0,0,-1]
[84,0,2,null,null,null,84,11,2,2,0,10,null,84,"A",-1,2,2,0,0,-2]
[85,0,1,null,null,null,85,10,1,2,1,11,null,85,"B",-2,1,2,1,1,-1]
[86,1,2,null,null,null,86,11,2,2,1,12,null,86,"B",-1,2,2,1,1,0]
[87,0,1,null,null,null,87,12,0,1,0,11,null,87,"A",0,0,1,1,0,-1]
[88,0,0,null,null,null,88,11,2,2,0,10,null,88,"B",-1,2,2,1,1,0]
[89,1,1,null,null,null,89,10,1,2,1,11,null,89,"B",0,0,1,0,0,-1]
[90,0,0,null,null,null,90,11,2,2,0,10,null,90,"A",-1,2,2,1,0,-2]
[91,0,-1,null,null,null,91,10,1,2,1,11,null,91,"B",-2,1,2,1,1,-1]
[92,1,0,null,null,null,92,11,2,2,0,10,null,92,"B",-1,2,2,1,1,0]
[93,0,-1,null,null,null,93,10,1,2,1,11,null,93,"A",0,0,1,0,0,-1]
[94,1,0,null,null,null,94,11,2,2,1,12,null,94,"B",-1,2,2,1,1,0]
[95,1,1,null,null,null,95,12,0,1,0,11,null,95,"B",0,0,1,0,0,-1]
[96,1,2,null,null,null,96,11,2,2,1,12,null,96,"A",-1,2,2,1,1,0]
[97,1,3,null,null,null,97,12,{"$f":"MOD(H104,3)","v":0},{"$f":"IF(I104=0,1,2)","v":1},0,11,null,97,"B",0,{"$f":"MOD(P104,3)","v":0},1,0,{"$f":"IF(O104=\"A\",B104,S104)","v":0},-1]
[98,1,{"$f":"IF(B105=1,1+C104,-1+C104)","v":4},null,null,null,98,11,{"$f":"MOD(H105,3)","v":2},{"$f":"IF(I105=0,1,2)","v":2},1,{"$f":"IF(K105=1,1+L104,-1+L104)","v":12},null,98,"B",-1,{"$f":"MOD(P105,3)","v":2},{"$f":"IF(Q105=0,1,2)","v":2},0,{"$f":"IF(O105=\"A\",B105,S105)","v":0},{"$f":"IF(T105=1,1+U104,-1+U104)","v":-2}]
[99,1,{"$f":"IF(B106=1,1+C105,-1+C105)","v":5},null,null,null,99,12,{"$f":"MOD(H106,3)","v":0},{"$f":"IF(I106=0,1,2)","v":1},0,{"$f":"IF(K106=1,1+L105,-1+L105)","v":11},null,99,"A",-2,{"$f":"MOD(P106,3)","v":1},{"$f":"IF(Q106=0,1,2)","v":2},0,{"$f":"IF(O106=\"A\",B106,S106)","v":1},{"$f":"IF(T106=1,1+U105,-1+U105)","v":-1}]
[100,0,{"$f":"IF(B107=1,1+C106,-1+C106)","v":4},null,null,null,100,11,{"$f":"MOD(H107,3)","v":2},{"$f":"IF(I107=0,1,2)","v":2},0,{"$f":"IF(K107=1,1+L106,-1+L106)","v":10},null,100,"B",-1,{"$f":"MOD(P107,3)","v":2},{"$f":"IF(Q107=0,1,2)","v":2},1,{"$f":"IF(O107=\"A\",B107,S107)","v":1},{"$f":"IF(T107=1,1+U106,-1+U106)","v":0}]

{"sheet":"MCParrOpt","index":7,"n_rows":104,"n_cols":14,"rows_kept":104,"cut":false,"hidden":true,"merged":[[2,4,2,6]]}
["Analytical Solution via Markov Chain"]
[null,null,null,null,null,null,null,null,"Initial M",0]
["epsilon",5.0000000000000001E-3,null,null,"Mod(M,3)"]
["Transition Matrix: Game A",null,null,"Game",0,1,2,null,"Flip #","Expected M"]
[null,null,null,"A",{"$f":"IF(MOD($J$2,3)=0,1,0)","v":1},{"$f":"IF(MOD($J$2,3)=1,1,0)","v":0},{"$f":"IF(MOD($J$2,3)=2,1,0)","v":0},null,1,{"$f":"IF(D5=\"A\",E5*($B$6-$C$6)+F5*($C$7-$A$7)+G5*($A$8-$B$8),E5*($B$12-$C$12)+F5*($C$13-$A$13)+G5*($A$14-$B$14))+J2","v":-1.0000000000000009E-2}]
[0,{"$f":"0.5-B3","v":0.495},{"$f":"0.5+B3","v":0.505},"B",{"$f":"IF(D5=\"A\",MMULT(E5:G5,$A$6:$C$8),MMULT(E5:G5,$A$12:$C$14))","v":0},0.495,0.505,null,2,{"$f":"IF(D6=\"A\",E6*($B$6-$C$6)+F6*($C$7-$A$7)+G6*($A$8-$B$8),E6*($B$12-$C$12)+F6*($C$13-$A$13)+G6*($A$14-$B$14))+J5","v":0.48}]
[{"$f":"0.5+B3","v":0.505},0,{"$f":"0.5-B3","v":0.495},"A",{"$f":"IF(D6=\"A\",MMULT(E6:G6,$A$6:$C$8),MMULT(E6:G6,$A$12:$C$14))","v":0.50244999999999995},0.128775,0.36877500000000002,null,3,{"$f":"IF(D7=\"A\",E7*($B$6-$C$6)+F7*($C$7-$A$7)+G7*($A$8-$B$8),E7*($B$12-$C$12)+F7*($C$13-$A$13)+G7*($A$14-$B$14))+J6","v":0.47}]
[{"$f":"0.5-B3","v":0.495},{"$f":"0.5+B3","v":0.505},0,"B",{"$f":"IF(D7=\"A\",MMULT(E7:G7,$A$6:$C$8),MMULT(E7:G7,$A$12:$C$14))","v":0.24757500000000002},0.43494412500000001,0.317480875,null,4,0.6381524999999999]
[null,null,null,"B",{"$f":"IF(D8=\"A\",MMULT(E8:G8,$A$6:$C$8),MMULT(E8:G8,$A$12:$C$14))","v":0.34743400375},0.10447724812500001,0.54808874812500008,null,5,0.67648829512499997]
["Transition Matrix: Game B",null,null,"A",{"$f":"IF(D9=\"A\",MMULT(E9:G9,$A$6:$C$8),MMULT(E9:G9,$A$12:$C$14))","v":0.43496781562500009},0.17276886112812503,0.39226332324687502,null,6,0.66648829512499996]
[null,null,null,"B",{"$f":"IF(D10=\"A\",MMULT(E10:G10,$A$6:$C$8),MMULT(E10:G10,$A$12:$C$14))","v":0.28141861987690625},0.41340204697404692,0.30517933314904694,null,7,0.79064408928502183]
[0,{"$f":"0.1-B3","v":9.5000000000000001E-2},{"$f":"0.9+B3","v":0.90500000000000003},"B",{"$f":"IF(D11=\"A\",MMULT(E11:G11,$A$6:$C$8),MMULT(E11:G11,$A$12:$C$14))","v":0.3327761251744219},0.10455549884131307,0.56266837598426522,null,8,0.84803512655827351]
[{"$f":"0.25+B3","v":0.255},0,{"$f":"0.75-B3","v":0.745},"A",{"$f":"IF(D12=\"A\",MMULT(E12:G12,$A$6:$C$8),MMULT(E12:G12,$A$12:$C$14))","v":0.4458495923128124},0.17509416776755771,0.37905623991963006,null,9,0.8380351265582735]
[{"$f":"0.75-B3","v":0.745},{"$f":"0.25+B3","v":0.255},0,"B",{"$f":"IF(D13=\"A\",MMULT(E13:G13,$A$6:$C$8),MMULT(E13:G13,$A$12:$C$14))","v":0.27605539348283353},0.4121189493542553,0.31182565716291133,null,10,0.96916311503058994]
[null,null,null,"B",{"$f":"IF(D14=\"A\",MMULT(E14:G14,$A$6:$C$8),MMULT(E14:G14,$A$12:$C$14))","v":0.33740044667170405},0.10574080495741157,0.55685874837088456,null,11,1.0205425343573746]
[null,null,null,"A",{"$f":"IF(D15=\"A\",MMULT(E15:G15,$A$6:$C$8),MMULT(E15:G15,$A$12:$C$14))","v":0.44182367280044893},0.17405202326838745,0.38412430393116381,null,12,1.0105425343573746]
[null,null,null,"B",{"$f":"IF(D16=\"A\",MMULT(E16:G16,$A$6:$C$8),MMULT(E16:G16,$A$12:$C$14))","v":0.27803780219646174},0.41268549152145995,0.30927670628207848,null,13,1.1390933915019743]
[null,null,null,"B",{"$f":"IF(D17=\"A\",MMULT(E17:G17,$A$6:$C$8),MMULT(E17:G17,$A$12:$C$14))","v":0.33564594651812074},0.10527915131059389,0.55907490217128553,null,14,1.1927536610284173]
[null,null,null,"A",{"$f":"IF(D18=\"A\",MMULT(E18:G18,$A$6:$C$8),MMULT(E18:G18,$A$12:$C$14))","v":0.44335698570180915},0.17445046497289929,0.38219254932529168,null,15,1.1827536610284173]
[null,null,null,"B",{"$f":"IF(D19=\"A\",MMULT(E19:G19,$A$6:$C$8),MMULT(E19:G19,$A$12:$C$14))","v":0.27728279672733352},0.41246894533166784,0.31024825794099875,null,16,1.3122860252828836]
[null,null,null,"B",{"$f":"IF(D20=\"A\",MMULT(E20:G20,$A$6:$C$8),MMULT(E20:G20,$A$12:$C$14))","v":0.33631453322561938},0.10545517146405137,0.55823029531032931,null,17,1.3650771320895785,null,null,"Final M",{"$f":"J104","v":6.1362268993107882}]
[null,null,null,"A",{"$f":"IF(D21=\"A\",MMULT(E21:G21,$A$6:$C$8),MMULT(E21:G21,$A$12:$C$14))","v":0.44277263872952843},0.17429860596056782,0.38292875530990378,null,18,1.3550771320895785]
[null,null,null,"B",{"$f":"IF(D22=\"A\",MMULT(E22:G22,$A$6:$C$8),MMULT(E22:G22,$A$12:$C$14))","v":0.27757052988848913},0.41255147760261801,0.30987799250889292,null,19,1.4842354432345426]
[null,null,null,"B",{"$f":"IF(D23=\"A\",MMULT(E23:G23,$A$6:$C$8),MMULT(E23:G23,$A$12:$C$14))","v":0.33605973120779281},0.10538808842917416,0.55855218036303311,null,20,1.537357792664412]
[null,null,null,"A",{"$f":"IF(D24=\"A\",MMULT(E24:G24,$A$6:$C$8),MMULT(E24:G24,$A$12:$C$14))","v":0.44299533691989906},0.17435648045731375,0.38264818262278721,null,21,1.527357792664412]
[null,null,null,"B",{"$f":"IF(D25=\"A\",MMULT(E25:G25,$A$6:$C$8),MMULT(E25:G25,$A$12:$C$14))","v":0.27746087302922312},0.41252002399985754,0.31001910297091934,null,22,1.656658657726422]
[null,null,null,"B",{"$f":"IF(D26=\"A\",MMULT(E26:G26,$A$6:$C$8),MMULT(E26:G26,$A$12:$C$14))","v":0.33615683783329858},0.10541365419536064,0.55842950797134083,null,23,1.7096547685431338]
[null,null,null,"A",{"$f":"IF(D27=\"A\",MMULT(E27:G27,$A$6:$C$8),MMULT(E27:G27,$A$12:$C$14))","v":0.44291046525846589},0.17433442412685529,0.38275511061467893,null,24,1.6996547685431338]
[null,null,null,"B",{"$f":"IF(D28=\"A\",MMULT(E28:G28,$A$6:$C$8),MMULT(E28:G28,$A$12:$C$14))","v":0.27750266393832801},0.41253201116335347,0.30996532489831863,null,25,1.8289013054233074]
[null,null,null,"B",{"$f":"IF(D29=\"A\",MMULT(E29:G29,$A$6:$C$8),MMULT(E29:G29,$A$12:$C$14))","v":0.3361198298959025},0.10540391092321241,0.55847625918088517,null,26,1.8819455265586342]
[null,null,null,"A",{"$f":"IF(D30=\"A\",MMULT(E30:G30,$A$6:$C$8),MMULT(E30:G30,$A$12:$C$14))","v":0.44294281037517863},0.17434282993123645,0.38271435969358503,null,27,1.8719455265586342]
[null,null,null,"B",{"$f":"IF(D31=\"A\",MMULT(E31:G31,$A$6:$C$8),MMULT(E31:G31,$A$12:$C$14))","v":0.27748673716359901},0.41252744278097386,0.30998582005542724,null,28,2.0012127682459555]
[null,null,null,"B",{"$f":"IF(D32=\"A\",MMULT(E32:G32,$A$6:$C$8),MMULT(E32:G32,$A$12:$C$14))","v":0.33613393385044166},0.10540762414467586,0.55845844200488259,null,29,2.0542386542403812]
[null,null,null,"A",{"$f":"IF(D33=\"A\",MMULT(E33:G33,$A$6:$C$8),MMULT(E33:G33,$A$12:$C$14))","v":0.44293048345052988},0.17433962642703701,0.38272989012243325,null,30,2.044238654240381]
[null,null,null,"B",{"$f":"IF(D34=\"A\",MMULT(E34:G34,$A$6:$C$8),MMULT(E34:G34,$A$12:$C$14))","v":0.27749280695625811},0.41252918381984105,0.30997800922390095,null,31,2.1734980051972457]
[null,null,null,"B",{"$f":"IF(D35=\"A\",MMULT(E35:G35,$A$6:$C$8),MMULT(E35:G35,$A$12:$C$14))","v":0.33612855874586567},0.10540620901293926,0.55846523224119515,null,32,2.2265308788276204]
[null,null,null,"A",{"$f":"IF(D36=\"A\",MMULT(E36:G36,$A$6:$C$8),MMULT(E36:G36,$A$12:$C$14))","v":0.44293518131798987},0.17434084730236199,0.3827239713796482,null,33,2.2165308788276201]
[null,null,null,"B",{"$f":"IF(D37=\"A\",MMULT(E37:G37,$A$6:$C$8),MMULT(E37:G37,$A$12:$C$14))","v":0.27749049372061863},0.41252852029912734,0.30998098598025403,null,34,2.3457932369908159]
[null,null,null,"B",{"$f":"IF(D38=\"A\",MMULT(E38:G38,$A$6:$C$8),MMULT(E38:G38,$A$12:$C$14))","v":0.33613060723156674},0.10540674832842355,0.55846264444000981,null,35,2.3988234475897792]
[null,null,null,"A",{"$f":"IF(D39=\"A\",MMULT(E39:G39,$A$6:$C$8),MMULT(E39:G39,$A$12:$C$14))","v":0.44293339093155532},0.17434038201920135,0.38272622704924342,null,36,2.3888234475897789]
[null,null,null,"B",{"$f":"IF(D40=\"A\",MMULT(E40:G40,$A$6:$C$8),MMULT(E40:G40,$A$12:$C$14))","v":0.27749137530907214},0.41252877317098779,0.30997985151994012,null,37,2.5180846596879851]
[null,null,null,"B",{"$f":"IF(D41=\"A\",MMULT(E41:G41,$A$6:$C$8),MMULT(E41:G41,$A$12:$C$14))","v":0.33612982654095724},0.10540654279194658,0.55846363066709626,null,38,2.5711158851847409]
[null,null,null,"A",{"$f":"IF(D42=\"A\",MMULT(E42:G42,$A$6:$C$8),MMULT(E42:G42,$A$12:$C$14))","v":0.44293407325893308},0.17434055934150047,0.38272536739956653,null,39,2.5611158851847406]
[null,null,null,"B",{"$f":"IF(D43=\"A\",MMULT(E43:G43,$A$6:$C$8),MMULT(E43:G43,$A$12:$C$14))","v":0.27749103933024316},0.41252867679995298,0.30998028386980392,null,40,2.6903775340554246]
[null,null,null,"B",{"$f":"IF(D44=\"A\",MMULT(E44:G44,$A$6:$C$8),MMULT(E44:G44,$A$12:$C$14))","v":0.33613012406699194},0.1054066211231731,0.55846325480983505,null,41,2.7434083727683349]
[null,null,null,"A",{"$f":"IF(D45=\"A\",MMULT(E45:G45,$A$6:$C$8),MMULT(E45:G45,$A$12:$C$14))","v":0.44293381321973629},0.1743404917628722,0.38272569501739168,null,42,2.7334083727683347]
[null,null,null,"B",{"$f":"IF(D46=\"A\",MMULT(E46:G46,$A$6:$C$8),MMULT(E46:G46,$A$12:$C$14))","v":0.27749116737385932},0.41252871352755227,0.30998011909858858,null,43,2.8626698551823178]
[null,null,null,"B",{"$f":"IF(D47=\"A\",MMULT(E47:G47,$A$6:$C$8),MMULT(E47:G47,$A$12:$C$14))","v":0.33613001067797432},0.10540659127065671,0.55846339805136913,null,44,2.9157008413009513]
[null,null,null,"A",{"$f":"IF(D48=\"A\",MMULT(E48:G48,$A$6:$C$8),MMULT(E48:G48,$A$12:$C$14))","v":0.44293391232228746},0.17434051751750668,0.38272557016020603,null,45,2.9057008413009511]
[null,null,null,"B",{"$f":"IF(D49=\"A\",MMULT(E49:G49,$A$6:$C$8),MMULT(E49:G49,$A$12:$C$14))","v":0.27749111857564285},0.41252869953043636,0.30998018189392096,null,46,3.0349623871526155]
[null,null,null,"B",{"$f":"IF(D50=\"A\",MMULT(E50:G50,$A$6:$C$8),MMULT(E50:G50,$A$12:$C$14))","v":0.33613005389123241},0.10540660264763592,0.55846334346113191,null,47,3.0879933170940133]
[null,null,null,"A",{"$f":"IF(D51=\"A\",MMULT(E51:G51,$A$6:$C$8),MMULT(E51:G51,$A$12:$C$14))","v":0.44293387455369043},0.1743405077022557,0.38272561774405411,null,48,3.077993317094013]
[null,null,null,"B",{"$f":"IF(D52=\"A\",MMULT(E52:G52,$A$6:$C$8),MMULT(E52:G52,$A$12:$C$14))","v":0.27749113717294593},0.41252870486482407,0.30998015796223022,null,49,3.2072548387691833]
[null,null,null,"B",{"$f":"IF(D53=\"A\",MMULT(E53:G53,$A$6:$C$8),MMULT(E53:G53,$A$12:$C$14))","v":0.33613003742239167},0.10540659831179858,0.55846336426581,null,50,3.2602857901200744]
[null,null,null,"A",{"$f":"IF(D54=\"A\",MMULT(E54:G54,$A$6:$C$8),MMULT(E54:G54,$A$12:$C$14))","v":0.4429338889475371},0.17434051144290877,0.38272559960955443,null,51,3.2502857901200741]
[null,null,null,"B",{"$f":"IF(D55=\"A\",MMULT(E55:G55,$A$6:$C$8),MMULT(E55:G55,$A$12:$C$14))","v":0.27749113008539839},0.41252870283185583,0.30998016708274612,null,52,3.3795473210090563]
[null,null,null,"B",{"$f":"IF(D56=\"A\",MMULT(E56:G56,$A$6:$C$8),MMULT(E56:G56,$A$12:$C$14))","v":0.33613004369876909},0.10540659996421312,0.55846335633701805,null,53,3.4325782642006564]
[null,null,null,"A",{"$f":"IF(D57=\"A\",MMULT(E57:G57,$A$6:$C$8),MMULT(E57:G57,$A$12:$C$14))","v":0.44293388346195284},0.17434051001732268,0.38272560652072479,null,54,3.4225782642006561]
[null,null,null,"B",{"$f":"IF(D58=\"A\",MMULT(E58:G58,$A$6:$C$8),MMULT(E58:G58,$A$12:$C$14))","v":0.27749113278650672},0.41252870360663263,0.30998016360686093,null,55,3.5518397915781974]
[null,null,null,"B",{"$f":"IF(D59=\"A\",MMULT(E59:G59,$A$6:$C$8),MMULT(E59:G59,$A$12:$C$14))","v":0.33613004130680268},0.10540659933446767,0.55846335935872982,null,56,3.6048707378793541]
[null,null,null,"A",{"$f":"IF(D60=\"A\",MMULT(E60:G60,$A$6:$C$8),MMULT(E60:G60,$A$12:$C$14))","v":0.44293388555254298},0.17434051056062236,0.38272560388683485,null,57,3.5948707378793539]
[null,null,null,"B",{"$f":"IF(D61=\"A\",MMULT(E61:G61,$A$6:$C$8),MMULT(E61:G61,$A$12:$C$14))","v":0.27749113175709755},0.41252870331136038,0.30998016493154229,null,58,3.7241322665951273]
[null,null,null,"B",{"$f":"IF(D62=\"A\",MMULT(E62:G62,$A$6:$C$8),MMULT(E62:G62,$A$12:$C$14))","v":0.33613004221839593},0.10540659957446755,0.55846335820713677,null,59,3.7771632117112128]
[null,null,null,"A",{"$f":"IF(D63=\"A\",MMULT(E63:G63,$A$6:$C$8),MMULT(E63:G63,$A$12:$C$14))","v":0.44293388475580608},0.17434051035356751,0.38272560489062663,null,60,3.7671632117112126]
[null,null,null,"B",{"$f":"IF(D64=\"A\",MMULT(E64:G64,$A$6:$C$8),MMULT(E64:G64,$A$12:$C$14))","v":0.27749113214941179},0.41252870342389047,0.30998016442669801,null,61,3.8964247399169776]
[null,null,null,"B",{"$f":"IF(D65=\"A\",MMULT(E65:G65,$A$6:$C$8),MMULT(E65:G65,$A$12:$C$14))","v":0.33613004187098211},0.10540659948300211,0.55846335864601615,null,62,3.9494556854847009]
[null,null,null,"A",{"$f":"IF(D66=\"A\",MMULT(E66:G66,$A$6:$C$8),MMULT(E66:G66,$A$12:$C$14))","v":0.44293388505944753},0.17434051043247742,0.38272560450807536,null,63,3.9394556854847007]
[null,null,null,"B",{"$f":"IF(D67=\"A\",MMULT(E67:G67,$A$6:$C$8),MMULT(E67:G67,$A$12:$C$14))","v":0.27749113199989839},0.41252870338100456,0.30998016461909733,null,64,4.0687172138848329]
[null,null,null,"B",{"$f":"IF(D68=\"A\",MMULT(E68:G68,$A$6:$C$8),MMULT(E68:G68,$A$12:$C$14))","v":0.3361300420033837},0.10540659951786016,0.55846335847875639,null,65,4.1217481592804344]
[null,null,null,"A",{"$f":"IF(D69=\"A\",MMULT(E69:G69,$A$6:$C$8),MMULT(E69:G69,$A$12:$C$14))","v":0.44293388494372787},0.17434051040240434,0.38272560465386807,null,66,4.1117481592804346]
[null,null,null,"B",{"$f":"IF(D70=\"A\",MMULT(E70:G70,$A$6:$C$8),MMULT(E70:G70,$A$12:$C$14))","v":0.27749113205687892},0.41252870339734865,0.30998016454577271,null,67,{"$f":"IF(D71=\"A\",E71*($B$6-$C$6)+F71*($C$7-$A$7)+G71*($A$8-$B$8),E71*($B$12-$C$12)+F71*($C$13-$A$13)+G71*($A$14-$B$14))+J70","v":4.2410096876064918}]
[null,null,null,"B",{"$f":"IF(D71=\"A\",MMULT(E71:G71,$A$6:$C$8),MMULT(E71:G71,$A$12:$C$14))","v":0.33613004195292456},0.10540659950457554,0.55846335854250018,null,68,4.2940406330676897]
[null,null,null,"A",{"$f":"IF(D72=\"A\",MMULT(E72:G72,$A$6:$C$8),MMULT(E72:G72,$A$12:$C$14))","v":0.44293388498782937},0.17434051041386539,0.38272560459830551,null,69,4.2840406330676899]
[null,null,null,"B",{"$f":"IF(D73=\"A\",MMULT(E73:G73,$A$6:$C$8),MMULT(E73:G73,$A$12:$C$14))","v":0.27749113203516323},0.4125287033911198,0.30998016457371719,null,70,4.4133021614219778]
[null,null,null,"B",{"$f":"IF(D74=\"A\",MMULT(E74:G74,$A$6:$C$8),MMULT(E74:G74,$A$12:$C$14))","v":0.33613004197215485},0.10540659950963839,0.55846335851820694,null,71,4.4663331068581762]
[null,null,null,"A",{"$f":"IF(D75=\"A\",MMULT(E75:G75,$A$6:$C$8),MMULT(E75:G75,$A$12:$C$14))","v":0.44293388497102198},0.1743405104094975,0.38272560461948074,null,72,4.4563331068581764]
[null,null,null,"B",{"$f":"IF(D76=\"A\",MMULT(E76:G76,$A$6:$C$8),MMULT(E76:G76,$A$12:$C$14))","v":0.27749113204343923},0.41252870339349368,0.30998016456306737,null,73,4.5855946352017058]
[null,null,null,"B",{"$f":"IF(D77=\"A\",MMULT(E77:G77,$A$6:$C$8),MMULT(E77:G77,$A$12:$C$14))","v":0.3361300419648261},0.10540659950770891,0.55846335852746531,null,74,4.6386255806474317]
[null,null,null,"A",{"$f":"IF(D78=\"A\",MMULT(E78:G78,$A$6:$C$8),MMULT(E78:G78,$A$12:$C$14))","v":0.44293388497742742},0.17434051041116214,0.38272560461141081,null,75,4.6286255806474319]
[null,null,null,"B",{"$f":"IF(D79=\"A\",MMULT(E79:G79,$A$6:$C$8),MMULT(E79:G79,$A$12:$C$14))","v":0.27749113204028519},0.41252870339258907,0.30998016456712607,null,76,4.7578871089950612]
[null,null,null,"B",{"$f":"IF(D80=\"A\",MMULT(E80:G80,$A$6:$C$8),MMULT(E80:G80,$A$12:$C$14))","v":0.33613004196761914},0.10540659950844423,0.55846335852393691,null,77,4.8109180544371561]
[null,null,null,"A",{"$f":"IF(D81=\"A\",MMULT(E81:G81,$A$6:$C$8),MMULT(E81:G81,$A$12:$C$14))","v":0.44293388497498626},0.17434051041052773,0.38272560461448629,null,78,4.8009180544371564]
[null,null,null,"B",{"$f":"IF(D82=\"A\",MMULT(E82:G82,$A$6:$C$8),MMULT(E82:G82,$A$12:$C$14))","v":0.27749113204148723},0.41252870339293379,0.30998016456557931,null,79,4.9301795827832233]
[null,null,null,"B",{"$f":"IF(D83=\"A\",MMULT(E83:G83,$A$6:$C$8),MMULT(E83:G83,$A$12:$C$14))","v":0.33613004196655472},0.10540659950816401,0.55846335852528162,null,80,4.9832105282267021]
[null,null,null,"A",{"$f":"IF(D84=\"A\",MMULT(E84:G84,$A$6:$C$8),MMULT(E84:G84,$A$12:$C$14))","v":0.44293388497591663},0.17434051041076951,0.38272560461331417,null,81,4.9732105282267023]
[null,null,null,"B",{"$f":"IF(D85=\"A\",MMULT(E85:G85,$A$6:$C$8),MMULT(E85:G85,$A$12:$C$14))","v":0.2774911320410291},0.4125287033928024,0.30998016456616878,null,82,5.1024720565733643]
[null,null,null,"B",{"$f":"IF(D86=\"A\",MMULT(E86:G86,$A$6:$C$8),MMULT(E86:G86,$A$12:$C$14))","v":0.33613004196696039},0.1054065995082708,0.55846335852476914,null,83,5.1555030020163164]
[null,null,null,"A",{"$f":"IF(D87=\"A\",MMULT(E87:G87,$A$6:$C$8),MMULT(E87:G87,$A$12:$C$14))","v":0.44293388497556202},0.17434051041067736,0.38272560461376093,null,84,5.1455030020163166]
[null,null,null,"B",{"$f":"IF(D88=\"A\",MMULT(E88:G88,$A$6:$C$8),MMULT(E88:G88,$A$12:$C$14))","v":0.27749113204120374},0.41252870339285247,0.30998016456594413,null,85,5.2747645303627522]
[null,null,null,"B",{"$f":"IF(D89=\"A\",MMULT(E89:G89,$A$6:$C$8),MMULT(E89:G89,$A$12:$C$14))","v":0.33613004196680574},0.10540659950823011,0.55846335852496454,null,86,5.327795475805905]
[null,null,null,"A",{"$f":"IF(D90=\"A\",MMULT(E90:G90,$A$6:$C$8),MMULT(E90:G90,$A$12:$C$14))","v":0.44293388497569725},0.1743405104107125,0.38272560461359062,null,87,5.3177954758059052]
[null,null,null,"B",{"$f":"IF(D91=\"A\",MMULT(E91:G91,$A$6:$C$8),MMULT(E91:G91,$A$12:$C$14))","v":0.27749113204113718},0.41252870339283343,0.30998016456602984,null,88,5.4470570041524269]
[null,null,null,"B",{"$f":"IF(D92=\"A\",MMULT(E92:G92,$A$6:$C$8),MMULT(E92:G92,$A$12:$C$14))","v":0.33613004196686475},0.10540659950824563,0.55846335852489004,null,89,5.5000879495955033]
[null,null,null,"A",{"$f":"IF(D93=\"A\",MMULT(E93:G93,$A$6:$C$8),MMULT(E93:G93,$A$12:$C$14))","v":0.44293388497564573},0.17434051041069912,0.38272560461365562,null,90,5.4900879495955035]
[null,null,null,"B",{"$f":"IF(D94=\"A\",MMULT(E94:G94,$A$6:$C$8),MMULT(E94:G94,$A$12:$C$14))","v":0.2774911320411626},0.41252870339284076,0.30998016456599714,null,91,5.6193494779419924]
[null,null,null,"B",{"$f":"IF(D95=\"A\",MMULT(E95:G95,$A$6:$C$8),MMULT(E95:G95,$A$12:$C$14))","v":0.33613004196684226},0.10540659950823972,0.55846335852491857,null,92,5.6723804233850981]
[null,null,null,"A",{"$f":"IF(D96=\"A\",MMULT(E96:G96,$A$6:$C$8),MMULT(E96:G96,$A$12:$C$14))","v":0.44293388497566544},0.17434051041070425,0.38272560461363081,null,93,5.6623804233850983]
[null,null,null,"B",{"$f":"IF(D97=\"A\",MMULT(E97:G97,$A$6:$C$8),MMULT(E97:G97,$A$12:$C$14))","v":0.27749113204115289},0.41252870339283798,0.30998016456600963,null,94,5.7916419517315996]
[null,null,null,"B",{"$f":"IF(D98=\"A\",MMULT(E98:G98,$A$6:$C$8),MMULT(E98:G98,$A$12:$C$14))","v":0.33613004196685087},0.10540659950824198,0.55846335852490769,null,95,5.8446728971746937]
[null,null,null,"A",{"$f":"IF(D99=\"A\",MMULT(E99:G99,$A$6:$C$8),MMULT(E99:G99,$A$12:$C$14))","v":0.44293388497565794},0.17434051041070231,0.3827256046136403,null,96,5.834672897174694]
[null,null,null,"B",{"$f":"IF(D100=\"A\",MMULT(E100:G100,$A$6:$C$8),MMULT(E100:G100,$A$12:$C$14))","v":0.27749113204115661},0.41252870339283904,0.30998016456600491,null,97,5.9639344255211908]
[null,null,null,"B",{"$f":"IF(D101=\"A\",MMULT(E101:G101,$A$6:$C$8),MMULT(E101:G101,$A$12:$C$14))","v":0.33613004196684759},0.10540659950824113,0.55846335852491191,null,98,6.0169653709642894]
[null,null,null,"A",{"$f":"IF(D102=\"A\",MMULT(E102:G102,$A$6:$C$8),MMULT(E102:G102,$A$12:$C$14))","v":0.44293388497566089},0.17434051041070306,0.38272560461363669,null,99,6.0069653709642896]
[null,null,null,"B",{"$f":"IF(D103=\"A\",MMULT(E103:G103,$A$6:$C$8),MMULT(E103:G103,$A$12:$C$14))","v":0.27749113204115522},0.41252870339283865,0.30998016456600674,null,100,6.1362268993107882,"Final M"]