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Lottery tickets are famously poor investments, but the mathematics is subtler than that slogan suggests. Under rare circumstances, a jackpot can grow large enough to give a ticket a positive expected value. Even then, however, the bet may remain irrational for an individual. The distinction reveals both the power of expected value and the limits of reducing a life-changing gamble to a single average.

When a Losing Ticket Becomes a “Good” Bet

Expected value combines every possible outcome with its probability. A wager that costs \$1 and pays \$100 for correctly predicting a fair die roll has an expected profit of almost \$16: the one-in-six chance of winning \$100 outweighs the five-in-six chance of losing the stake. That does not mean a player will receive \$16. The actual result is either a \$100 win or a \$1 loss. The figure describes the average return per play if the same wager could be repeated many times.

Applying this logic to Powerball initially looks devastating. A ticket costs \$2, the jackpot begins around \$20 million, and the chance of matching all six numbers is one in 292,201,338. Considering only the jackpot, the article calculates an expected value of about -\$1.93 per ticket. Even that is generous. The advertised jackpot assumes a 29-year annuity rather than the smaller lump-sum payment, federal and possibly state taxes reduce the payout, and a complete calculation must include the game’s smaller prizes.

Rollover jackpots seem capable of changing the answer. The odds of matching the numbers and the ticket price remain fixed while the prize grows, so at some point the simple expected-value calculation can become positive. Yet the crowds drawn by giant jackpots introduce a complication: the winning prize may have to be divided among multiple tickets.

The Crowd Changes the Odds

The first U.S. lottery jackpot to exceed \$1 billion illustrates the problem. A buying frenzy pushed the January 2016 Powerball prize to \$1.56 billion and produced sales of more than 635 million tickets—over 20 times the average drawing that year. With so many combinations in circulation, the probability that more than one ticket would win exceeded 60 percent. Three tickets did win and split the jackpot. After incorporating the likelihood of shared prizes, taxes and partial-match awards, even that record-setting drawing still had negative expected value.

Ticket choices are not perfectly random, either. Many people select birthdays and anniversaries, concentrating their picks below 31. They also favor odd numbers and avoid multiples of 10, perhaps because those choices look more random. No selection can improve the probability that a ticket’s numbers are drawn, but avoiding popular patterns can reduce the chance of sharing a jackpot. Large even numbers and multiples of 10 may therefore improve the payout conditional on winning.

Surprisingly, the two largest U.S. jackpots at the time of the article—in November 2022 and October 2023—apparently attracted few enough buyers to produce positive expected values even after adjustments for taxes and prize splitting. Smaller state lotteries may occasionally offer similar opportunities because they generate less attention. These cases are rare, however, and players generally cannot identify them in advance because final ticket-sales totals are not published before a drawing. Finding a mathematically favorable lottery is itself a gamble.

Why Positive Expected Value Is Not Enough

Expected value works best when a person can make a moderate wager repeatedly and allow averages to emerge. A lottery jackpot is the opposite: an extreme payoff with odds so remote that no individual can play enough times for the long-run average to become a realistic personal outcome. A positive value on paper therefore does not imply that buying many tickets is safe or sensible.

The calculation also treats every dollar as equally valuable. In practice, the first \$50 million would transform a winner’s life far more than the next \$50 million. Economists describe this as diminishing marginal utility. Expected value likewise ignores risk aversion: most people feel a loss more sharply than an equal-sized gain, and a person should not stake money needed for rent, food or emergencies merely because an abstract average is favorable.

That does not make every ticket indefensible. A buyer may be purchasing a few days of pleasurable anticipation rather than an investment, much as someone pays for another form of entertainment. Lottery revenue can also support public services such as education, and some research suggests that the anticipation itself can lift a player’s mood regardless of the result.

The mathematical lesson is therefore not simply “never play.” Expected value clarifies when the numbers are favorable, but it cannot decide whether a rare, volatile payoff is worth pursuing or how much entertainment is worth to a particular person. A lottery can occasionally qualify as a good bet in the narrow mathematical sense while remaining a bad financial decision—and sometimes a modestly priced fantasy can be understood without pretending it is an investment.