Kelly criterion
You have $25 and a coin that is biased in favor of heads 60% of the time. Your friend gives you 1 to 1 odds, a $1 bet could win an additional $1 (on top of getting back your $1 wagered), for a bet on heads. How much of money should you risk on your first flip, so that you make the most money?
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1 solution
https://en.m.wikipedia.org/wiki/Kelly_criterion
In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet is a formula used to determine the optimal size of a series of bets in order to maximize wealth. f ∗ = b b p − q = b b p − ( 1 − p ) = b p ( b + 1 ) − 1
where:
f * is the fraction of the current bankroll to wager, i.e. how much to bet;
b is the net odds received on the wager ("b to 1"); that is, you could win $b (on top of getting back your $1 wagered) for a $1 bet
p is the probability of winning;
q is the probability of losing, which is 1 − p.