Is a Lottery Ticket Ever Worth It? Expected Value

The math of a lottery ticket is simple: multiply every prize by its chance of hitting, add them all up, and subtract the ticket price. That's the lottery expected value. For a $2 Powerball ticket, that number is almost always negative. Compute it for yourself using the official prize table and adjust for the real-world factors that change the answer.

The formula is Σ (prize × probability) for every tier, then subtract the cost of the ticket. An expected value of -$1.50 means you lose $1.50 per $2 ticket on average over millions of plays. The long-run average is not what happens on one ticket, you either lose $2 or win something, but it is the number that tells you whether the game is rigged in your favor.

Tier-by-Tier: The Powerball Prize Table

Powerball publishes fixed prizes for every non-jackpot tier. The jackpot is pari-mutuel, meaning the advertised amount is not what one winner gets; it is the annuity value, and the cash option is roughly 60% of that. The table below uses the official odds from the Powerball prize and odds page.

Use the cash option, not the annuity. A $500 million advertised jackpot pays roughly $300 million cash. Assume one winner for now.

Jackpot contribution: $300,000,000 × (1/292,201,338) ≈ $1.03. That is the biggest single term. Now add every other tier. The $1 million prize: $1,000,000 × (1/11,688,053.52) ≈ $0.09. The $50,000 prize: $50,000 × (1/913,129.18) ≈ $0.05. The $100 prizes: two tiers, each at about $100 × (1/36,525.17) and $100 × (1/14,494.11) ≈ $0.003 and $0.007. The $7 prizes: two tiers, sum to about $0.02. The $4 prizes: two tiers, sum to about $0.17.

Sum of all non-jackpot prizes: about $0.34. Total EV before ticket cost: $1.03 + $0.34 = $1.37. Subtract the $2 ticket price: EV = -$0.63. You lose 63 cents per $2 play on average.

This is the naive EV. It assumes one winner, no tax, and a cash option that tracks the advertised jackpot at 60%. Each of those assumptions is wrong in the real world.

The advertised jackpot is an annuity paid over 30 years. The cash option, what you actually get at once, is roughly 60% of that, though the exact percentage varies with interest rates. The Powerball and Mega Millions prize and odds pages both state that the cash option is the standard payout for most winners.

The IRS withholds 24% on prizes over $5,000, per Publication 505 (2025), Table 4. State taxes add 0% to 8.82%. Using a $500 million jackpot with a $300 million cash option: after federal withholding you get $228 million. After a 5% state tax (a middle estimate), you get about $216 million. Recalculate the jackpot term: $216,000,000 × (1/292,201,338) ≈ $0.74. Total EV: $0.74 + $0.34 - $2 = -$0.92. You lose 92 cents per ticket.

The gap between the advertised jackpot and the after-tax cash EV is roughly 40-60%, as the Claimed vs. Real data shows. For Mega Millions, the same adjustment applies: the jackpot odds are 1 in 302,575,350, and the cash option is similarly discounted.

The expected value of a lottery ticket falls as more tickets are sold because the jackpot is split among winners. The expected number of winners is roughly 1 + (tickets sold × probability of winning). If 300 million tickets are sold for a Powerball drawing, the expected number of winners is 1 + (300,000,000 / 292,201,338) ≈ 2.03. The jackpot term must be divided by that number.

Using the after-tax cash jackpot of $216 million from the example above: $216,000,000 / 2.03 ≈ $106.4 million. Jackpot term: $106.4 million × (1/292,201,338) ≈ $0.36. Total EV: $0.36 + $0.34 - $2 = -$1.30. That is a 58% drop from the naive EV.

This is the core reason that bigger jackpots do not make the lottery worth it. More media coverage means more ticket sales, which means more expected winners, which pushes the EV further negative. The Powerball drawing frequency is three times per week; Mega Millions draws twice. A rollover that grows the jackpot also grows the pool of buyers.

The naive EV, no split, no tax, annuity value, turns positive when the jackpot exceeds roughly $585 million for Powerball. That number comes from setting the jackpot term equal to the $2 ticket price minus the non-jackpot EV ($0.34): ($1.66) / (1/292,201,338) = $485 million, then adding back the 40% annuity discount to get $808 million advertised. Using the cash option directly: $2 - $0.34 = $1.66 needed from the jackpot. $1.66 × 292,201,338 = $485 million cash. At 60% of advertised, that is $808 million advertised.

But that number is useless. Add the split risk: even at a $1.5 billion advertised jackpot, if 400 million tickets are sold, the expected number of winners is 1.37. The after-tax EV per ticket is still negative by roughly $0.50. The IRS withholding rate, state taxes, and the cash discount mean that the break-even jackpot is probably above $2 billion, and even then only if ticket sales are low, which they never are at that level.

The Wizard of Odds house-edge tables show that casino games like blackjack (basic strategy) have a house edge of 0.5% to 2.0%. A lottery ticket's house edge, by contrast, is around 40% to 60% at most jackpot levels. That is not a game you should play for profit.

High-school and college statistics students who need to compute EV from a probability table will find the Powerball structure a perfect textbook problem. Use a TI-84 Plus CE to compute EV and standard deviation using 1-Var Stats with a frequency list. Use Microsoft Excel or Google Sheets with the SUMPRODUCT function to multiply prize and probability arrays. The OpenStax Introductory Statistics 2e textbook covers discrete random variables in Chapter 4, and the Blitzstein & Hwang Introduction to Probability (2nd edition) covers linearity of expectation in Chapter 4.

Project managers who use Expected Monetary Value (EMV) in decision trees, as described in the PMBOK Guide (7th edition), can apply the same multiplication-and-sum method to risk registers. The Kelly Criterion, from the 1956 Bell System Technical Journal, provides the next step: how much of a bankroll to bet when you have a known edge.

If you are trying to decide whether to buy a single lottery ticket for fun, the EV calculation tells you that you are paying for the entertainment of a tiny chance at a huge prize, not for a financial edge. If you are looking for a guaranteed outcome on one trial, this is not the right tool. The probability of losing the entire $2 is 96%, and the probability of winning the jackpot is effectively zero. The single thing that most often goes wrong is treating a positive naive EV at a high jackpot as a signal to buy many tickets. It is not. The after-tax, split-adjusted EV is still negative, and variance is extreme, one standard deviation on a $2 ticket can be hundreds of dollars, meaning a long losing streak will exhaust any reasonable bankroll.