The Expected Value of a Lottery Ticket, Explained

What expected value means

Expected value is the average outcome if you repeated the same bet millions of times. For a lottery ticket, it answers the question: if you bought this exact ticket at this exact price with this exact jackpot an infinite number of times, what would you get back on average?

The formula is simple. For each possible outcome, multiply the prize by the probability of winning it. Add all those products together. That is the expected value of the ticket.

If the expected value is $1.50 and the ticket costs $2, you lose $0.50 per ticket on average. The expected value is negative.

Powerball expected value at different jackpot sizes

A $2 Powerball ticket has 9 possible winning outcomes, from the $4 minimum prize (matching just the Powerball) up to the jackpot. At a $20 million starting jackpot, here is how the math works:

The jackpot contributes the most to expected value. $20 million times 1/292,201,338 equals about $0.068. Less than seven cents.

The $1 million second prize (5 numbers, no Powerball) at 1 in 11,688,053 adds about $0.086.

The $50,000 third prize at 1 in 913,129 adds about $0.055.

All the smaller prizes ($4 to $100) combined contribute roughly $0.32.

Total expected value: approximately $0.53. You pay $2 for a ticket worth $0.53 in mathematical terms. You lose $1.47 per ticket on average.

As the jackpot grows, the expected value rises. At $500 million, the jackpot contribution jumps to about $1.71, pushing total expected value to roughly $2.22. That looks positive, but there are two problems.

Why a "positive" expected value is a mirage

First, taxes. A $500 million jackpot becomes roughly $173 million after lump sum reduction and federal taxes. The expected value of the jackpot component drops from $1.71 to about $0.59 when you use the after-tax number.

Second, splitting. When jackpots get large, ticket sales spike. More tickets sold means higher probability of multiple winners splitting the prize. At a $500 million jackpot, ticket sales typically double or triple compared to a $50 million jackpot. The probability of splitting can cut the expected value of the jackpot in half or more.

After adjusting for taxes and split probability, the expected value of a Powerball ticket is negative at every jackpot size that has ever existed. The break-even point (where expected value equals $2) requires a jackpot that has never been reached after adjustments.

Mega Millions expected value

Mega Millions has slightly worse base odds (1 in 302,575,350 for the jackpot) and a $2 ticket price. The math plays out similarly.

At a $20 million jackpot, expected value is about $0.49. At $500 million, it rises to about $2.10 before adjustments and drops to about $0.85 after taxes and split probability.

The "any prize" odds are 1 in 24 for Mega Millions versus 1 in 24.9 for Powerball, so the small-prize contribution is slightly better. But the jackpot is where most of the expected value lives, and the odds are worse.

Scratch-off expected value

Scratch-off tickets have published return rates. A typical $1 ticket returns about 60 cents per dollar spent. A typical $20 ticket returns about 72 cents per dollar.

The expected value calculation for scratch-offs is more straightforward because the prizes are fixed (no growing jackpot). A $20 ticket with a 72% return rate has an expected value of $14.40. You pay $20, you get $14.40 back on average. You lose $5.60 per ticket.

The wrinkle with scratch-offs is that the expected value changes as prizes are claimed. If the top prize of a game has already been won, the expected value of remaining tickets drops. Most state lottery websites publish which prizes remain, so you can recalculate.

EuroMillions and UK Lotto expected value

EuroMillions tickets cost 2.50 euros. The jackpot odds are 1 in 139,838,160. At a 17 million euro starting jackpot, the expected value is roughly 0.80 euros. You lose 1.70 euros per ticket.

UK Lotto tickets cost 2 pounds. Jackpot odds are 1 in 45,057,474. At a 2 million pound minimum jackpot, the expected value is approximately 0.90 pounds. But UK lottery winnings are tax-free, which changes the math significantly compared to US games.

With no tax on UK winnings, the expected value at high jackpots can actually approach the ticket price more closely than any US lottery. At a 50 million pound jackpot, the EV is roughly 1.85 pounds on a 2 pound ticket. Still negative, but much closer to break-even than Powerball ever gets after taxes.

What expected value does not capture

Expected value treats a $1 million prize as equal to a million $1 prizes. In mathematical terms it is. In human terms it is not. A million dollars changes your life. A dollar does not.

Economists call this the "utility" distinction. The utility of winning $1 million is not a million times the utility of winning $1. For most people, the first $100,000 of a windfall is life-changing (pay off debt, emergency fund, maybe a down payment). The difference between $5 million and $6 million is barely noticeable in terms of quality of life.

This is why people play the lottery despite negative expected value. They are not buying expected value. They are buying a chance at a life-changing event. The $2 ticket price is low enough that losing it has near-zero impact on their daily life, but the potential prize has enormous impact. In utility terms, it can be rational to take a bet with negative expected value if the cost is negligible and the upside is transformative.

Whether that reasoning is correct or just a rationalization for a bad bet depends on who you ask. The math says it is always a losing proposition. Human experience says $2 for a few days of daydreaming about financial freedom is not the worst thing you could spend money on.

How to use expected value practically

If you are going to play, expected value can help you choose games. Higher return-rate scratch-offs have better expected value than lower ones. Games with tax-free winnings (UK Lotto, EuroMillions) have better after-tax expected value than US games. Smaller-format games (SA Lotto, Loto France) have better odds and better expected value per ticket than Powerball or Mega Millions.

None of them are positive expected value after all adjustments. But some are less negative than others. If you are going to spend money on lottery tickets regardless, the calculator can show you which games lose the least per dollar.