How this calculator works
Game A wins one unit with probability 0.5 - epsilon. Game B uses probability 0.1 - epsilon when capital is divisible by three and 0.75 - epsilon otherwise. A loss costs one unit; capital may become negative, with no absorbing bankruptcy rule. The calculator tracks probabilities of the three capital remainders and the expected reward each round. Random mixing changes the state distribution, so its gain is not a weighted average of the standalone long-run gains. Short runs and some mixtures need not exhibit the paradox.
Formula
p_A = 1/2 - epsilon. p_B(0) = 1/10 - epsilon; p_B(1) = p_B(2) = 3/4 - epsilon. Random mixture: p_r = gamma p_A + (1 - gamma)p_B(r). Expected one-step gain = sum_r probability(r)(2p_r - 1).
Worked example
With epsilon zero and starting capital zero, game A has first-step expected gain 0 and game B has gain -0.8. A 50:50 random mixture has gain -0.4 on that first step. With epsilon 0.005 over longer runs, mixing can reverse the standalone losses.
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Frequently asked questions
Does combining any two losing games make a winner?
No. The state dependence and switching rule are essential. Many combinations remain losing.
Can a player go bankrupt in this model?
Capital is allowed to take any integer value. A stopping rule at zero would change the state process and the expected gains.
Is the table one simulated path?
No. Each entry is an expected gain obtained by propagating probabilities. Individual play can differ greatly from the expectation.