When the lottery is genuinely +EV
Kelly staking is the standard answer to 'how much should I bet given an edge?'. Applied to a lottery ticket it returns a number so small that no human bankroll can act on it.
Suppose you have found a genuinely positive-expected-value lottery — a roll-down draw, say. Expected value tells you the bet is favourable. It does not tell you how much to stake. For that, the standard tool is the Kelly criterion, and its answer here is instructive.
John Kelly's 1956 result gives the fraction of your bankroll to stake in order to maximise the long-run growth rate of that bankroll. For a bet that wins with probability p and pays b-to-1:
f* = (bp − q) ÷ b, where q = 1 − p
The numerator (bp − q) is your edge in payout units; dividing by b scales it by how extreme the payout is. Two properties matter for what follows:
Take US Powerball: p = 1 ÷ 292,201,338. Assume a $2 ticket in a scenario generous enough to be genuinely positive — say the ticket is worth $2.60 in expectation, a 30% edge, which no real Powerball draw has ever offered.
The payout ratio b is enormous: winning a $500 million cash jackpot on a $2 ticket is b = 250,000,000-to-1.
f* = (bp − q) ÷ b
Force it favourable. Make the jackpot large enough that bp = 1.3 (a 30% edge). Then:
f* = (1.3 − 1) ÷ 250,000,000 = 0.3 ÷ 250,000,000 = 1.2 × 10⁻⁹
That is the fraction of your wealth Kelly permits: about one-billionth.
At a 30% edge — an edge no jackpot game has ever offered — growth-optimal staking says a billionaire may buy one ticket.
The intuition is worth having, because it generalises far beyond lotteries.
Kelly's denominator is the payout odds. A bet that pays 250 million to one has, by construction, a probability of winning that is essentially zero on any human timescale. Long-run growth is driven by what happens in the overwhelming majority of trials, and in the overwhelming majority of trials your ticket is worthless. The rare enormous payoff cannot compensate within any number of draws you will ever play.
Put differently: expected value averages over outcomes you will never sample. Kelly averages over outcomes weighted by how often you actually experience them. When those two diverge as violently as they do in a lottery, Kelly is the more useful number.
This is the formal version of the argument in why even a +EV jackpot isn't worth chasing.
Kelly is not anti-gambling. It permits large stakes on favourable bets with moderate odds:
| Bet | Edge | Payout odds | Kelly fraction |
|---|---|---|---|
| Coin flip paying 2:1 | 50% | 2:1 | 25% of bankroll |
| Card-counted blackjack hand | ~1% | ~1:1 | ~1% of bankroll |
| Roll-down lottery, lower divisions | ~10% | ~1,000:1 | ~0.01% of bankroll |
| Jackpot lottery at 30% edge | 30% | 250,000,000:1 | 0.0000001% |
Notice the third row. That is the shape of the Cash WinFall exploit: a modest edge on short-odds lower divisions, where Kelly permits a real stake, repeated over hundreds of thousands of tickets. The syndicates were not betting on the jackpot; they were grinding the reachable tiers (Cash WinFall). Kelly explains precisely why that was the right structure and jackpot-chasing is not.
For any lottery jackpot, at any jackpot size, for any bankroll a person could have, the growth-optimal stake is effectively zero.
That is not a moral position. It is what the formula returns, and the formula is the standard tool for exactly this question.
Which leaves one legitimate framing for buying a ticket, and it is not an investment framing: entertainment spending. A ticket costs a couple of dollars and buys a few days of daydream, and if that is worth the money to you, it is worth the money — see the cost of hope and setting a lottery budget you won't regret.
Just do not confuse it with a bet you are supposed to size.
Last verified: 2026-08-29