A positive-EV lottery you'd barely touch

Question

A counterparty offers this bet: for each $\$ 1 $you stake, with probability$ 0.1% $you receive$ $2{,}000 $(a$ 2000\times $payoff), and otherwise you lose the stake. The expected payoff per dollar is$ 0.001 \times 2000 = $2 $, a net expected profit of$ $1$ per dollar staked. It looks like free money.

Would you bet the whole book? How much should a growth-maximizing desk actually risk, and why?

Accepted Answer

The EV is genuinely positive. Stake \1, expect \2 back, net +\1. Maximizing expected profit alone says bet everything you can. That is exactly the trap the St. Petersburg paradox warns about: a huge mean carried by a tiny-probability outcome is nothing like a steady edge, because you almost never see the good outcome. What actually happens if you bet big. With a 0.1\% hit rate, you lose your entire stake 999 times out of 1000. Bet the whole book and you are essentially certain to be wiped out before the rare jackpot arrives. Expected wealth is maximized by betting everything, but median wealt…

A positive-EV lottery you'd barely touch

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