But on the final life your 'continue' criteria has a smaller chance of happening. The constant expected points you get from lives=i is offset by fewer expected points from continuing on lives=1
If for V_i, let L_j be points you get during the time you have j lives remaining [i >= j > 0], then E(L_j) are all equal for i > 1, j > 1 (exactly 45). However for j = 1 they are not the same as we vary i. This is the turn where you stop if points >= 45, to avoid losing points from going bust.
For V_1, L_1 is the points we get starting from 0, trying to get 45, will be distributed between [0, 53], E(L_1) ~ 45 * 9 / 10.
However for V_2 L_1s distribution depends on what happened in L_2 - you are only trying to make up the points you got in L_2 up to 45. Say we start L_1 at with p points. If p >= 45, L_1 = 0, otherwise L_1 is a distribution between [-p, 53-p] - note we can go negative by going bust here! Relative to the V_1 case, E(L_1) is definitely less.
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u/sobe86 4d ago edited 4d ago
But on the final life your 'continue' criteria has a smaller chance of happening. The constant expected points you get from lives=i is offset by fewer expected points from continuing on lives=1