Positive Expected Value in a Lottery: When It's Real and When It Isn't

Every time a jackpot rolls past half a billion, someone points out that the advertised prize divided by the odds now exceeds the ticket price, and concludes the lottery has become a good bet. The arithmetic is right. The model is missing three terms, and each of them is large.

The naive calculation

Expected value per ticket is the sum, over every division, of (prize × probability), minus the ticket price:

EV = Σ (prize × P(prize)) − price

Take US Powerball at an advertised $1 billion. Jackpot probability is 1 in 292,201,338, so:

$1,000,000,000 ÷ 292,201,338 = $3.42 of jackpot value per ticket

Powerball's non-jackpot prizes are fixed, so they can be summed exactly:

So EV ≈ $3.42 + $0.32 = $3.74 against a $2 ticket. Nearly double your money. Buy everything.

Except that $3.42 does not exist.

Killer 1: the advertised jackpot is not a sum of money

The headline figure is an annuity — what you receive if the operator invests a smaller sum and pays you instalments over decades. The amount actually available today is the cash value, which in recent US jackpots has run roughly 45–55% of the advertised figure.

Call it 50%: $500,000,000.

Jackpot contribution: $500,000,000 ÷ 292,201,338 = $1.71

Running EV: $1.71 + $0.32 = $2.03. Still marginally above the $2 ticket. Two adjustments to go. (If you would rather take the annuity, you are accepting the operator's discount rate — see annuity vs lump sum.)

Killer 2: tax

In the United States a jackpot is ordinary income. Federal withholding is 24% immediately, and the top marginal rate of 37% applies on a prize this size when the return is filed; most states add their own. Take 37% federal alone and ignore state tax entirely:

$500,000,000 × 0.63 = $315,000,000 $315,000,000 ÷ 292,201,338 = $1.08

Running EV: $1.08 + $0.32 = $1.40 against a $2 ticket. Now underwater.

Tax is jurisdiction-specific and not universal — UK and Australian prizes are paid tax-free, which is why the same arithmetic looks different there. The country-by-country table has the rates, and the after-tax prize calculator applies them.

Killer 3: jackpot sharing — the one everybody forgets

This is the largest term and the least intuitive. Jackpots are shared between all winning tickets. The bigger the jackpot, the more tickets are sold, and the more likely someone else holds your combination.

If S tickets are sold and your jackpot probability is p, the number of other winning tickets is well modelled as Poisson with mean λ = S × p. Your expected share of the jackpot is:

(1 − e^(−λ)) ÷ λ

At a billion-dollar Powerball, US sales have run in the hundreds of millions of tickets. Take 600 million:

λ = 600,000,000 ÷ 292,201,338 = 2.05 Share factor = (1 − e^(−2.05)) ÷ 2.05 = (1 − 0.1287) ÷ 2.05 = 0.425

You expect to keep about 42.5% of the jackpot you win. Apply it:

$315,000,000 × 0.425 = $134,000,000 $134,000,000 ÷ 292,201,338 = $0.46

One consistency point, because it is easy to get wrong: the $0.32 of fixed prizes computed earlier was a pre-tax figure. Those prizes are taxable income too, so to add them to a post-tax jackpot they must be taxed as well:

$0.32 × 0.63 = $0.20

Final EV: $0.46 + $0.20 = $0.66 against a $2 ticket.

The ticket returns about 33 cents on the dollar at the largest jackpot in the game's history. (If you assume the small prizes escape tax in practice, the figure is $0.78 — about 39 cents on the dollar. The conclusion does not move.) The naive calculation said $3.74. The gap between those numbers is the entire subject of this article.

The cruel feedback loop

Notice what makes killer 3 vicious: sales rise with the jackpot. The very growth that makes the prize look attractive is what recruits the co-winners who dilute it. The jackpot and the sharing penalty climb together, and the sharing penalty eventually wins.

This is why the break-even jackpot for a large game is often not merely high but unreachable: by the time the jackpot got there, sales would have risen enough to push break-even higher still. Chasing it is chasing a horizon.

So when is it genuinely positive?

Real +EV situations have existed. They share a common shape: a structural rule that redistributes money into the lower divisions, where sharing is mild and prizes are modest.

• Roll-downs. When a capped jackpot goes unwon, some games push it down into lower tiers. This is what made Massachusetts Cash WinFall exploitable — the definitive account is in the Cash WinFall roll-down, and the general mechanism in roll-down mechanics.
• Mispriced structures. The 1729 Paris bond lottery paid prizes disproportionate to ticket cost — see the Voltaire syndicate.
• Instant-game inventory. A scratch game late in its life can be +EV if the big prizes are unclaimed; the scratch card EV calculator does that arithmetic.

Every one of those is a rules exploit, not a jackpot-size exploit. Modern jackpot games are structured so no jackpot level makes them positive.

Even if it is positive, that is not a reason to play

Positive expected value and worth doing are different claims. A bet with a 1-in-292-million payoff has variance so enormous that the optimal stake for any human bankroll rounds to zero — see the Kelly criterion applied to a lottery ticket and why even a +EV jackpot usually isn't worth chasing.

Test any game and jackpot yourself with the expected value calculator — it applies all three adjustments.