How lottery odds work
In a draw lottery, you pick a set of numbers from a pool, often plus a bonus ball from a second pool. The number of possible tickets is the number of ways to choose those numbers, a combination:
Combinations = C(n, k) × C(m, b)
where C(n, k) = n! ÷ (k! × (n − k)!) is the number of ways to choose k numbers from n. Your chance of winning the jackpot with one ticket is 1 divided by that number.
The big US games
| Game | Format | Jackpot odds | Ticket |
|---|---|---|---|
| Powerball | 5 of 69 + 1 of 26 | 1 in 292,201,338 | $2 |
| Mega Millions | 5 of 70 + 1 of 24 | 1 in 290,472,336 | $5 |
Mega Millions changed its format in April 2025, with fewer Mega Balls and a higher ticket price.
Worked example: Powerball
- C(69, 5) = 11,238,513 ways to pick the white balls
- × 26 Powerballs = 292,201,338 combinations
- Probability per ticket ≈ 0.000000342%
- Tickets needed for a 50% chance of winning at least once: about 202.5 million, costing about $405 million
- Buying one ticket every second, it would take 9.3 years to cover every combination
Is a ticket worth it?
The expected value of the jackpot is the prize times the probability. For a $100 million advertised jackpot, that is $100,000,000 ÷ 292,201,338 ≈ $0.34 per $2 ticket, before considering:
- the cash option, typically around half of the advertised annuity value
- federal tax of up to 37% plus state tax in most states
- the chance of sharing the jackpot with other winners
Smaller prizes add some value, but the overall return of US lotteries is roughly 50 to 60 cents per dollar. Even record jackpots rarely make a ticket a positive-value bet.
Myths
- Hot and cold numbers: each draw is independent; past results do not change future odds.
- Quick picks versus your own numbers: the odds are identical. Avoiding popular patterns, such as birthdays (1 to 31) or sequences, only reduces the risk of splitting a prize.
- Buying more tickets: each ticket adds the same tiny chance. Ten tickets give 10 chances in 292 million.
Play for fun
Treat a lottery ticket as entertainment with a known cost, not as an investment. If gambling stops being fun, call 1-800-GAMBLER for free, confidential help.