Kelly criterion & expected value calculator

Expected value of a bet

Expected value (EV) is the average result per unit staked if you repeated the bet many times. With win probability p and a payout multiplier m (decimal odds, stake included), EV per $1 = p × m − 1. A 55% chance at odds of 1.95 gives 0.55 × 1.95 − 1 = +0.0725, a 7.25% edge. A 49.5% dice roll paying 2× gives 0.99 − 1 = −0.01, the 1% house edge.

The implied probability of a payout is 1 ÷ m. If your honest estimate of the win chance is higher than the implied probability, the bet has positive EV.

The Kelly criterion formula

Kelly picks the stake that maximises the long-run growth rate of your bankroll. With net odds b = m − 1 and loss probability q = 1 − p, the Kelly fraction is f* = (b × p − q) ÷ b, which simplifies to (p × m − 1) ÷ (m − 1). For the 55% at 1.95 example that is 0.0725 ÷ 0.95 = 7.63% of your current bankroll.

When EV is zero or negative the formula returns zero or a negative number, which means do not bet. Every casino original, slot and table game has a built-in house edge, so Kelly's answer for all of them is 0%, no matter which multiplier you choose.

Full Kelly, half Kelly and flat betting

The chart plots median bankroll paths. For a fraction f of bankroll per bet, the expected log growth per bet is g = p × ln(1 + f × b) + q × ln(1 − f), so the median bankroll after n bets is B × e^(n × g). Flat betting a fixed dollar amount grows linearly at stake × EV per bet.

Full Kelly grows fastest in the long run but has deep drawdowns: there is a 50% chance of falling to half your bankroll at some point. Half Kelly gives up about a quarter of the growth rate in exchange for much smaller swings, and it protects you if your edge estimate is too optimistic. Betting more than twice the Kelly fraction gives negative growth even with a real edge. Press Simulate a run to see one random sequence of outcomes applied to all three strategies.

Where positive EV actually exists

Sports bettors find +EV when their probability model beats the bookmaker's line. In a casino, the only levers are external: rakeback, deposit bonuses with light wagering, and promotions. Those reduce your expected loss but rarely flip a game to positive EV on their own.