This is the story of expected gain, and stat. information for the Chow-Robbins coin-tossing game with life duration, 1000 for the first, 10, tosses with prob. of getting at least a certain amount with resolution, 0.1 The expected gain at the very beginning is, 0.7925637640 The standard deviation is, 0.2153004519 The prob. of getting at least, 0.5 + 0.1 i, for i from 1 to, 5 are: [[0.6, 0.6917238235], [0.7, 0.5625000000], [0.8, 0.5000000000], [0.9, 0.5000000000], [1.0, 0.5000000000]] If the total number of tosses is, 1, then whenever the number of heads minus the number of tails is >=, 1 in other words, the number of heads is >=, 1 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 1, tails is, 0.5851275280 The standard deviation is, 0.08154130946 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.3834476471], [0.7, 0.1250000000], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 2, then whenever the number of heads minus the number of tails is >=, 2 in other words, the number of heads is >=, 2 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 2, tails is, 0.5521920162 The standard deviation is, 0.05138151921 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.2247772217], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 1, tails is, 0.6180630399 The standard deviation is, 0.09213258872 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.5421180725], [0.7, 0.2500000000], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 3, then whenever the number of heads minus the number of tails is >=, 3 in other words, the number of heads is >=, 3 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 3, tails is, 0.5373889635 The standard deviation is, 0.03915226753 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.1130065918], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 2, tails is, 0.5669950689 The standard deviation is, 0.05752356251 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.3365478516], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 1, tails is, 0.6691310108 The standard deviation is, 0.09193471733 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.7476882935], [0.7, 0.5000000000], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 4, then whenever the number of heads minus the number of tails is >=, 2 in other words, the number of heads is >=, 3 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 4, tails is, 0.5290498109 The standard deviation is, 0.03221429700 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.04829406738], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 3, tails is, 0.5457281161 The standard deviation is, 0.04346212424 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.1777191162], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 2, tails is, 0.5882620217 The standard deviation is, 0.06184171478 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.4953765869], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 5, then whenever the number of heads minus the number of tails is >=, 3 in other words, the number of heads is >=, 4 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 5, tails is, 0.5236176550 The standard deviation is, 0.02717657916 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.01806640625], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 4, tails is, 0.5344819668 The standard deviation is, 0.03574826950 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.07852172852], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 3, tails is, 0.5569742654 The standard deviation is, 0.04740276331 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.2769165039], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 3, heads and , 2, tails is, 0.6195497780 The standard deviation is, 0.05868497266 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.7138366699], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 6, then whenever the number of heads minus the number of tails is >=, 2 in other words, the number of heads is >=, 4 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 6, tails is, 0.5197870180 The standard deviation is, 0.02364548939 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.005249023438], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 5, tails is, 0.5274482921 The standard deviation is, 0.02981067224 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.03088378906], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 4, tails is, 0.5415156415 The standard deviation is, 0.03960121466 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.1261596680], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 3, heads and , 3, tails is, 0.5724328892 The standard deviation is, 0.04947574791 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.4276733398], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 7, then whenever the number of heads minus the number of tails is >=, 3 in other words, the number of heads is >=, 5 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 7, tails is, 0.5169322129 The standard deviation is, 0.02095306410 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.001220703125], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 6, tails is, 0.5226418230 The standard deviation is, 0.02574660402 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.009277343750], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 5, tails is, 0.5322547611 The standard deviation is, 0.03268425387 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.05249023438], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 3, heads and , 4, tails is, 0.5507765219 The standard deviation is, 0.04355139657 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.1998291016], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 4, heads and , 3, tails is, 0.5940892565 The standard deviation is, 0.04539800248 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.6555175781], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 8, then whenever the number of heads minus the number of tails is >=, 2 in other words, the number of heads is >=, 5 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 8, tails is, 0.5147157712 The standard deviation is, 0.01880593357 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.0002441406250], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 7, tails is, 0.5191486546 The standard deviation is, 0.02268421092 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.002197265625], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 6, tails is, 0.5261349915 The standard deviation is, 0.02804990839 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.01635742188], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 3, heads and , 5, tails is, 0.5383745308 The standard deviation is, 0.03570462688 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.08862304688], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 4, heads and , 4, tails is, 0.5631785131 The standard deviation is, 0.04702136893 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.3110351562], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 9, then whenever the number of heads minus the number of tails is >=, 3 in other words, the number of heads is >=, 6 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 9, tails is, 0.5129418680 The standard deviation is, 0.01705829791 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 8, tails is, 0.5164896743 The standard deviation is, 0.02024962426 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.0004882812500], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 7, tails is, 0.5218076348 The standard deviation is, 0.02459591867 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.003906250000], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 3, heads and , 6, tails is, 0.5304623482 The standard deviation is, 0.03051529901 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.02880859375], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 4, heads and , 5, tails is, 0.5462867133 The standard deviation is, 0.03864261925 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.1484375000], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 5, heads and , 4, tails is, 0.5800703128 The standard deviation is, 0.04856027650 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.4736328125], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] If the total number of tosses is, 10, then whenever the number of heads minus the number of tails is >=, 4 in other words, the number of heads is >=, 7 it is a stop, and you collect for sure h/(h+t) for the remaining case we have The expected gain with, 0, heads and , 10, tails is, 0.5114882476 The standard deviation is, 0.01560954493 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 1, heads and , 9, tails is, 0.5143954884 The standard deviation is, 0.01827805072 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 2, heads and , 8, tails is, 0.5185838602 The standard deviation is, 0.02184573638 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.0009765625000], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 3, heads and , 7, tails is, 0.5250314095 The standard deviation is, 0.02668139402 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.006835937500], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 4, heads and , 6, tails is, 0.5358932869 The standard deviation is, 0.03303755394 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.05078125000], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 5, heads and , 5, tails is, 0.5566801398 The standard deviation is, 0.04097532730 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.2460937500], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The expected gain with, 6, heads and , 4, tails is, 0.6034604858 The standard deviation is, 0.04407973538 The prob. of getting at least .5+r*i, for i from 1 to, 5 are: [[0.6, 0.7011718750], [0.7, 0.], [0.8, 0.], [0.9, 0.], [1.0, 0.]] The whole thing took, 3154.301, secs. of CPU time