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Based on the computer-generated results, when played optimally, which is to avoid banking, your expectation at the beginning of the game is 38.1733 units.
However, simulations show that the probability of getting 8 Wins in a row during a play of 27 rolls is only a small fraction above 27%. That would seem to indicate that the expected value should be only about 128*.271 or 34.7. How does it get higher? I puzzled over this for a while and realized that if you haven't reached stage 8 when you get to the 27th roll, the program credits you with the amount for the stage in which you were correct, as if there is an automatic bank after the completion of the 27th roll. It made me add the rule stating that fact.
If it reaches the point of the $128,000 question, if you bank, you've got $64,000; if you don't bank, you have a 2/3 probability of getting $128,000 added to your winnings and 1/3 chance of getting nothing except a new beginning (with a reduced number of remaining questions). It seems clear it's worth avoiding banking.
Each column below represents a stage from 1 through 8; each row is a successive roll of the die.
38.1733 \\40.1451 43.0596 47.3672 53.8758 63.7705 78.6033 100.8392
36.8392 38.8403 \\41.7981 46.1697 52.6313 62.3941 77.2361 99.4854
35.4854 37.5162 40.5178 \\44.9544 51.5119 61.2042 75.8485 98.1115
34.1115 36.1724 39.2185 43.7210 \\50.3758 60.2121 74.7505 96.7170
32.7170 34.8087 37.9000 42.4693 49.2229 \\59.2052 73.9596 95.7673
31.3018 33.4247 36.5621 41.1991 48.0530 58.1835 \\73.1569 95.2885
29.8655 32.0200 35.2042 39.9104 46.8659 57.1467 72.3425 \\94.8027
28.4080 30.5943 33.8259 38.6024 45.6616 56.0949 71.5161 94.3097
26.9291 29.1474 32.4269 37.2743 44.4390 55.0279 70.6778 93.8096
25.4288 27.6793 31.0068 35.9259 43.1971 53.9441 69.8275 93.3023
23.9069 26.1897 29.5655 34.5567 41.9354 52.8423 68.9627 92.7878
22.3633 24.6786 28.1029 33.1666 40.6534 51.7215 68.0818 92.2624
20.7873 23.1513 26.6243 31.7607 39.3562 50.5865 67.1886 91.7290
19.1869 21.5875 25.1335 30.3430 38.0476 49.4409 66.2863 91.1894
17.5683 19.9962 23.5971 28.9162 36.7304 48.2872 65.3772 90.6453
15.9359 18.3844 22.0264 27.4277 35.4063 47.1276 64.4629 90.0978
14.2933 16.7572 20.4300 25.8930 33.9949 45.9628 63.5447 89.5477
12.6430 15.1185 18.8144 24.3236 32.5180 44.6709 62.6227 88.9956
10.9868 13.4710 17.1843 22.7281 30.9919 43.2835 61.5129 88.4407
9.3221 11.8192 15.5455 21.1154 29.4312 41.8269 60.2643 87.6083
6.8249 10.5706 14.3164 19.9058 28.2606 40.7343 59.3278 86.9840
4.9520 7.7613 13.3800 18.9986 27.3827 39.9150 58.6255 86.5158
3.5473 5.6543 9.8683 18.2963 26.7243 39.3004 58.0988 86.1646
2.4938 4.0741 7.2346 13.5556 26.1975 38.8395 57.7037 85.9012
1.7037 2.8889 5.2593 10.0000 19.4815 38.4444 57.4074 85.7037
1.1111 2.0000 3.7778 7.3333 14.4444 28.6667 57.1111 85.5556
0.6667 1.3333 2.6667 5.3333 10.6667 21.3333 42.6667 85.3333
The optimization program avoids banking (would be a 1) in all rolls in all possible stages. (The impossible stages are shown to the right of the backslashes; they represent rolls in a more advanced stage than is possible with the stage on the first roll being 1.)
sayBank =
0 \\0 0 0 0 0 0 1
0 0 \\0 0 0 0 0 1
0 0 0 \\0 0 0 0 1
0 0 0 0 \\0 0 0 1
0 0 0 0 0 \\0 0 0
0 0 0 0 0 0 \\0 0
0 0 0 0 0 0 0 \\0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Calculations were done from high to low in both roll number and stage to allow recursing to previously calculate values:
p0=2/3;
sayBank=zeros(27,8);
for i=1:8
winnings(28,i)=0;
end
for i=1:28
winnings(i,9)=128;
end
for q=27:-1:1
ppot=128; pot=64;
for lvl=8:-1:1
bval=pot+winnings(q+1,1); % value of banking
if q==27
nbval=p0*ppot; % value of not banking
else
nbval=p0*(winnings(q+1,lvl+1))+(1-p0)*winnings(q+1,1);
end
if bval>nbval % which is the better choice?
sayBank(q,lvl)=1;
winnings(q,lvl)=bval;
else
winnings(q,lvl)=nbval;
end
ppot=ppot/2; pot=ppot/2;
if pot<1
pot=0;
end
end
end
clc
format short
winnings
sayBank
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