This is a comparative study of two gambling methodologies for playing the Chow-Robbins Coin-Tossing game, where you toss a fair coin, and you are allowed to stop any time collecting, as payoff, the average number of heads. The first way is the usual one, maximizing the expected gain. The second way is that you have to match or exceed a certain goal, and wishes to maximize the prob. of making it. (e.g. if you need a certain score to get admitted to your college of choice, and you need your SAT score to be at least 1500, or you want to buy something that costs a certain amount, and if you get one penny less , you can't.) Below is the comparative study for the first, 6, tosses where we can only toss the coin up to, 200, times (if we make it so far, we collect 1/2 if the number of tails is larger than the number of heads). ------------------------------------------ At the very beginning if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.7916879464, ExpectedGain (SecondWay):, 0.6742902054 Prob.Exceeding Goal(FirstWay):, 0.6917238235, Prob.Exc.Goal(SecondWay):, 0.7753928313 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.7916879464, ExpectedGain (SecondWay):, 0.5787939263 Prob.Exceeding Goal(FirstWay):, 0.5625000000, Prob.Exc.Goal(SecondWay):, 0.6075176458 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.7916879464, ExpectedGain (SecondWay):, 0.5289752098 Prob.Exceeding Goal(FirstWay):, 0.5000000000, Prob.Exc.Goal(SecondWay):, 0.5362190123 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.7916879464, ExpectedGain (SecondWay):, 0.5008867384 Prob.Exceeding Goal(FirstWay):, 0.5000000000, Prob.Exc.Goal(SecondWay):, 0.5009852649 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.7916879464, ExpectedGain (SecondWay):, 0.5000000000 Prob.Exceeding Goal(FirstWay):, 0.5000000000, Prob.Exc.Goal(SecondWay):, 0.5000000000 --------------- If the total number of tosses so far is, 1, then things are as follows. ---------------- If the number of heads is, 0, and the number of tails is, 1 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5833758928, ExpectedGain (SecondWay):, 0.3485804107 Prob.Exceeding Goal(FirstWay):, 0.3834476471, Prob.Exc.Goal(SecondWay):, 0.5507856625 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5833758928, ExpectedGain (SecondWay):, 0.1575878526 Prob.Exceeding Goal(FirstWay):, 0.1250000000, Prob.Exc.Goal(SecondWay):, 0.2150352915 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5833758928, ExpectedGain (SecondWay):, 0.05795041961 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.07243802452 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5833758928, ExpectedGain (SecondWay):, 0.001773476764 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.001970529738 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5833758928, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. If the total number of tosses so far is, 2, then things are as follows. ---------------- If the number of heads is, 0, and the number of tails is, 2 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5499390913, ExpectedGain (SecondWay):, 0.2138787519 Prob.Exceeding Goal(FirstWay):, 0.2247772217, Prob.Exc.Goal(SecondWay):, 0.3534793090 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5499390913, ExpectedGain (SecondWay):, 0.04925715691 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.06953943040 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5499390913, ExpectedGain (SecondWay):, 0.004197813917 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.005247267396 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5499390913, ExpectedGain (SecondWay):, -5 0.3494688702 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -5 0.3882987447 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5499390913, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 1, and the number of tails is, 1 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.6168126942, ExpectedGain (SecondWay):, 0.4832820696 Prob.Exceeding Goal(FirstWay):, 0.5421180725, Prob.Exc.Goal(SecondWay):, 0.7480920160 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.6168126942, ExpectedGain (SecondWay):, 0.2659185483 Prob.Exceeding Goal(FirstWay):, 0.2500000000, Prob.Exc.Goal(SecondWay):, 0.3605311526 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.6168126942, ExpectedGain (SecondWay):, 0.1117030253 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.1396287816 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.6168126942, ExpectedGain (SecondWay):, 0.003543458839 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.003937176488 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.6168126942, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. If the total number of tosses so far is, 3, then things are as follows. ---------------- If the number of heads is, 0, and the number of tails is, 3 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5347185013, ExpectedGain (SecondWay):, 0.1278600313 Prob.Exceeding Goal(FirstWay):, 0.1130065918, Prob.Exc.Goal(SecondWay):, 0.2107745859 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5347185013, ExpectedGain (SecondWay):, 0.01402485700 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.01992165418 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5347185013, ExpectedGain (SecondWay):, 0.0003040813474 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.0003801016843 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5347185013, ExpectedGain (SecondWay):, -8 0.6886388012 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -8 0.7651542236 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5347185013, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 1, and the number of tails is, 2 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5651596814, ExpectedGain (SecondWay):, 0.2998974725 Prob.Exceeding Goal(FirstWay):, 0.3365478516, Prob.Exc.Goal(SecondWay):, 0.4961840321 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5651596814, ExpectedGain (SecondWay):, 0.08448945682 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.1191572066 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5651596814, ExpectedGain (SecondWay):, 0.008091546486 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.01011443311 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5651596814, ExpectedGain (SecondWay):, -5 0.6982491017 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -5 0.7758323352 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5651596814, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 2, and the number of tails is, 1 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.6684657071, ExpectedGain (SecondWay):, 0.6666666667 Prob.Exceeding Goal(FirstWay):, 0.7476882935, Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.6684657071, ExpectedGain (SecondWay):, 0.4473476397 Prob.Exceeding Goal(FirstWay):, 0.5000000000, Prob.Exc.Goal(SecondWay):, 0.6019050986 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.6684657071, ExpectedGain (SecondWay):, 0.2153145041 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.2691431302 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.6684657071, ExpectedGain (SecondWay):, 0.007079935187 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.007866594653 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.6684657071, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. If the total number of tosses so far is, 4, then things are as follows. ---------------- If the number of heads is, 0, and the number of tails is, 4 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5260490542, ExpectedGain (SecondWay):, 0.07848360333 Prob.Exceeding Goal(FirstWay):, 0.04829406738, Prob.Exc.Goal(SecondWay):, 0.1300668832 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5260490542, ExpectedGain (SecondWay):, 0.003766079766 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.005339092342 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5260490542, ExpectedGain (SecondWay):, 0.00002202705210 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.00002753381512 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5260490542, ExpectedGain (SecondWay):, -10 0.1356983236 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -10 0.1507759152 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5260490542, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 1, and the number of tails is, 3 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5433879484, ExpectedGain (SecondWay):, 0.1772364593 Prob.Exceeding Goal(FirstWay):, 0.1777191162, Prob.Exc.Goal(SecondWay):, 0.2914822886 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5433879484, ExpectedGain (SecondWay):, 0.02428363423 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.03450421601 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5433879484, ExpectedGain (SecondWay):, 0.0005861356427 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.0007326695534 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5433879484, ExpectedGain (SecondWay):, -7 0.1375920619 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -7 0.1528800688 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5433879484, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 2, and the number of tails is, 2 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5869314143, ExpectedGain (SecondWay):, 0.4225584857 Prob.Exceeding Goal(FirstWay):, 0.4953765869, Prob.Exc.Goal(SecondWay):, 0.7008857755 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5869314143, ExpectedGain (SecondWay):, 0.1446952794 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.2038101972 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5869314143, ExpectedGain (SecondWay):, 0.01559695733 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.01949619666 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5869314143, ExpectedGain (SecondWay):, 0.00001395122283 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.00001550135870 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5869314143, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 3, and the number of tails is, 1 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.7500000000, ExpectedGain (SecondWay):, 0.7500000000 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.7500000000, ExpectedGain (SecondWay):, 0.7500000000 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.7500000000, ExpectedGain (SecondWay):, 0.4150320509 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.5187900637 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.7500000000, ExpectedGain (SecondWay):, 0.01414591915 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.01571768795 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.7500000000, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. If the total number of tosses so far is, 5, then things are as follows. ---------------- If the number of heads is, 0, and the number of tails is, 5 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5203577628, ExpectedGain (SecondWay):, 0.04761125939 Prob.Exceeding Goal(FirstWay):, 0.01806640625, Prob.Exc.Goal(SecondWay):, 0.07894074025 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5203577628, ExpectedGain (SecondWay):, 0.001106535185 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.001573879879 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5203577628, ExpectedGain (SecondWay):, -5 0.1595596140 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -5 0.1994495175 10 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5203577628, ExpectedGain (SecondWay):, -13 0.2673975821 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -13 0.2971084245 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5203577628, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 1, and the number of tails is, 4 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5317403456, ExpectedGain (SecondWay):, 0.1093559473 Prob.Exceeding Goal(FirstWay):, 0.07852172852, Prob.Exc.Goal(SecondWay):, 0.1811930261 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5317403456, ExpectedGain (SecondWay):, 0.006425624348 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.009104304806 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5317403456, ExpectedGain (SecondWay):, 0.00004245850806 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.00005307313507 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5317403456, ExpectedGain (SecondWay):, -10 0.2711292497 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -10 0.3012547219 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5317403456, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 2, and the number of tails is, 3 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5550355512, ExpectedGain (SecondWay):, 0.2451169714 Prob.Exceeding Goal(FirstWay):, 0.2769165039, Prob.Exc.Goal(SecondWay):, 0.4017715511 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5550355512, ExpectedGain (SecondWay):, 0.04214164412 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.05990412721 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5550355512, ExpectedGain (SecondWay):, 0.001129812777 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.001412265972 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5550355512, ExpectedGain (SecondWay):, -7 0.2749129946 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -7 0.3054588829 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5550355512, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 3, and the number of tails is, 2 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.6188272773, ExpectedGain (SecondWay):, 0.6000000000 Prob.Exceeding Goal(FirstWay):, 0.7138366699, Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.6188272773, ExpectedGain (SecondWay):, 0.2472489147 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.3477162673 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.6188272773, ExpectedGain (SecondWay):, 0.03006410188 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.03758012735 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.6188272773, ExpectedGain (SecondWay):, 0.00002787495436 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.00003097217151 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.6188272773, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 4, and the number of tails is, 1 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.8000000000, ExpectedGain (SecondWay):, 0.8000000000 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.8000000000, ExpectedGain (SecondWay):, 0.8000000000 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.8000000000, ExpectedGain (SecondWay):, 0.8000000000 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.8000000000, ExpectedGain (SecondWay):, 0.02826396335 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.03140440372 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.8000000000, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. If the total number of tosses so far is, 6, then things are as follows. ---------------- If the number of heads is, 0, and the number of tails is, 6 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5163180487, ExpectedGain (SecondWay):, 0.02907249128 Prob.Exceeding Goal(FirstWay):, 0.005249023438, Prob.Exc.Goal(SecondWay):, 0.04826745389 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5163180487, ExpectedGain (SecondWay):, 0.0003107633993 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.0004422717896 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5163180487, ExpectedGain (SecondWay):, -6 0.1155818323 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -6 0.1444772904 10 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5163180487, ExpectedGain (SecondWay):, -16 0.5269148873 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -16 0.5854609859 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5163180487, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 1, and the number of tails is, 5 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5243974769, ExpectedGain (SecondWay):, 0.06615002749 Prob.Exceeding Goal(FirstWay):, 0.03088378906, Prob.Exc.Goal(SecondWay):, 0.1096140266 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5243974769, ExpectedGain (SecondWay):, 0.001902306970 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.002705487968 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5243974769, ExpectedGain (SecondWay):, -5 0.3075610448 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -5 0.3844513060 10 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5243974769, ExpectedGain (SecondWay):, -13 0.5342682493 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -13 0.5936313881 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5243974769, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 2, and the number of tails is, 4 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5390832144, ExpectedGain (SecondWay):, 0.1525618670 Prob.Exceeding Goal(FirstWay):, 0.1261596680, Prob.Exc.Goal(SecondWay):, 0.2527720257 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5390832144, ExpectedGain (SecondWay):, 0.01094894173 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.01550312164 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5390832144, ExpectedGain (SecondWay):, 0.00008184140567 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.0001023017571 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5390832144, ExpectedGain (SecondWay):, -10 0.5417242311 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -10 0.6019158124 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5390832144, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 3, and the number of tails is, 3 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.5709878880, ExpectedGain (SecondWay):, 0.3376720757 Prob.Exceeding Goal(FirstWay):, 0.4276733398, Prob.Exc.Goal(SecondWay):, 0.5507710765 if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.5709878880, ExpectedGain (SecondWay):, 0.07333434651 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.1043051328 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.5709878880, ExpectedGain (SecondWay):, 0.002177784149 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.002722230186 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.5709878880, ExpectedGain (SecondWay):, -7 0.5492842650 10 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, -7 0.6103158500 10 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.5709878880, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 4, and the number of tails is, 2 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.6666666667, ExpectedGain (SecondWay):, 0.6666666667 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.6666666667, ExpectedGain (SecondWay):, 0.4211634829 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.5911274018 if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.6666666667, ExpectedGain (SecondWay):, 0.05795041961 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.07243802452 if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.6666666667, ExpectedGain (SecondWay):, 0.00005569498029 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.00006188331143 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.6666666667, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ---------------- If the number of heads is, 5, and the number of tails is, 1 if you want to exceed the goal of , 0.6 ExpectedGain (FirstWay):, 0.8333333333, ExpectedGain (SecondWay):, 0.8333333333 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.7 ExpectedGain (FirstWay):, 0.8333333333, ExpectedGain (SecondWay):, 0.8333333333 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.8 ExpectedGain (FirstWay):, 0.8333333333, ExpectedGain (SecondWay):, 0.8333333333 Prob.Exceeding Goal(FirstWay):, 1., Prob.Exc.Goal(SecondWay):, 1. if you want to exceed the goal of , 0.9 ExpectedGain (FirstWay):, 0.8333333333, ExpectedGain (SecondWay):, 0.05647223172 Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0.06274692413 if you want to exceed the goal of , 1.0 ExpectedGain (FirstWay):, 0.8333333333, ExpectedGain (SecondWay):, 0. Prob.Exceeding Goal(FirstWay):, 0., Prob.Exc.Goal(SecondWay):, 0. ------------------- ------------------- The whole thing took, 766.627, secs. of CPU time