P(ruin) theory
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Empirical ruin rate
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Current capital
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Fair game:
P(ruin) = 1 − k/N. Biased game (r = q/p, q = 1−p):
P(ruin) = (1 − r^k) / (1 − r^N). Each round the walk moves +1 with
probability p or −1 with probability q, stopping at the barriers 0 (ruin) or N (target).