Lottery odds: the complete guide

How lottery odds are calculated

Imagine a game where you pick 5 numbers from 1 to 69. How many different sets of 5 are there? Order does not matter, so we count combinations: 69 choose 5, which is 69×68×67×66×65 divided by 5×4×3×2×1, or 11,238,513. If the game also draws one bonus ball from 26, every main combination can pair with 26 bonus numbers, giving 11,238,513 × 26 = 292,201,338 possible tickets. Exactly one is the jackpot ticket, so the odds are 1 in 292,201,338.

Smaller prizes use the same idea. To win a prize for matching 4 numbers and the bonus ball, you count the number of tickets that match exactly 4 of the 5 drawn numbers (choosing 4 of the 5 winning numbers and 1 of the 64 others) and the bonus ball, and divide by the total. That is how you get 1 in 913,129 for that tier in Powerball.

You can see every tier for any game in our lottery odds calculator, which does this maths for you.

The odds of 14 major games

Computed from each game's published format, sorted from best to worst. Formats change from time to time, and some games have several prize levels and different ways of selling plays, so check the official rules. The key observation: the odds of the headline prize range from about 1 in 14 million to more than 1 in 600 million, and those with the biggest jackpots have the worst odds, because the jackpot is funded by sales and sales rise when the prize looks large.

Putting a big number into perspective

Try the how long to win the lottery calculator to see the years for your game and ticket count.

• If every person in a stadium of 70,000 held a Powerball ticket, you would need over 4,000 stadiums to cover every combination.
• A person is far more likely to be struck by lightning in their lifetime (commonly cited at about 1 in 15,300) than to win the Powerball jackpot with one ticket.
• If you bought one Powerball ticket every draw, three times a week, you would expect to wait more than a million years for a 50% chance of a jackpot.
• The best US jackpot odds in our table, Lotto America, are about 1 in 26 million, a far shorter shot but still tiny.

Do more tickets improve your odds?

Yes, in proportion. If every ticket is different, your chance of the jackpot is the number of tickets divided by the number of possible tickets. Ten Powerball tickets give you 10 in 292,201,338, which is a tiny number multiplied by ten. To have a 1% chance you would need about 2.9 million different tickets, costing about $5.8 million at $2 each. To have a 50% chance you would need about 146 million tickets, costing $292 million. That is more than the cash value of most jackpots.

Random quick picks can repeat, so the chance with n quick picks is 1 minus (1 minus 1/N) to the power n, a hair lower than n/N, which makes no practical difference for small n.

Quick pick, lucky numbers and past results

Every combination is equally likely to be drawn. A quick pick is as good as a set chosen from birthdays, and a set made of the numbers that came up last week is no better or worse than any other. Past results do not matter, because each draw is independent: the balls have no memory. Frequency tables of "hot" and "cold" numbers describe the past and have no predictive power. See our guide to lottery myths and the explanation of the gambler's fallacy.

One real effect exists, and it concerns prizes rather than odds. When several people pick the winning combination, they share the jackpot. Combinations made of numbers up to 31 (dates) are over-represented, so choosing numbers across the full range can reduce how often you would share, without changing your chance of winning. Our number generator can avoid date-only sets for you.

Expected value: what a ticket is worth on average

Expected value (EV) multiplies each possible prize by its probability and adds them up. A ticket's EV is what it returns on average, which you compare with its price. In a pool lottery all prizes come from sales, so EV per ticket equals the payout share times the ticket price. With a 90% payout, a $5 ticket has an EV of $4.50, an average cost of 50 cents. With a 50% payout the average cost is $2.50.

Large jackpot games add complexity: the jackpot is paid at the cash value, taxed, and shared if several people win. After those adjustments, expected value is almost always below the ticket price even when the headline jackpot is enormous. Use the expected value calculator to test a game, and read our explanation of lottery expected value.

Odds in a pool lottery

In a pool lottery such as CryptoDrawz, your odds depend on the number of tickets sold, T. With 14 winners, the chance one ticket wins any prize is 14/T and the grand prize is 1/T. At 1,000 tickets, that is 1.4% for something and 0.1% for the top prize. At 100 tickets, 14% and 1%. At 10,000 tickets, 0.14% and 0.01%. Prizes scale up as T grows because the pool is 90% of sales, so the average return stays at 90 cents per dollar.

A small pool is a different kind of game from a giant jackpot: smaller prizes, but a chance you can actually feel. That does not make it a good investment, since expected value is still negative, but it is a more honest product.

Variance: why real results look so different from averages

Expected value is an average that real players almost never experience. Most tickets win nothing, a few win small prizes and very rarely one wins big. If you play for a year and win nothing, that is the normal outcome. If you win a jackpot, that says nothing about the game being good, it is variance. Averages only show up over millions of plays. This is why a lucky streak never proves a lottery is worth playing and an unlucky one never proves it is rigged: verification, not feelings, is how you check fairness.

Is there any smart way to play?

Not for your chance per ticket. The choices that do change things are about money and risk. Pick games with a high payout share, because they lose you less on average. Prefer smaller draws if you value the chance of winning something. Set a firm budget, because spending less is the only reliable way to lose less. Choose lotteries you can verify, because the biggest risk of all is that the game is not what it claims. Our guide on lottery strategy goes through each of these, including syndicates.

Free tools for lottery odds

• Lottery odds calculator for Powerball, Mega Millions, EuroMillions, UK Lotto and Lotto 6/49.
• Odds with multiple tickets, and how many tickets you would need for a given chance.
• How long it would take to win, in years, for any game.
• Expected value calculator for jackpot games.
• Lottery spending calculator and the lottery vs investing calculator.

Worked examples you can check

Simulating 10,000 players

Imagine 10,000 players each buying one ticket in a draw of 10,000 tickets with 14 winners. On average 14 of them win something, 1 wins the grand prize, 3 win a runner-up prize and 10 win a lucky draw. But the number who win in any actual draw varies. Sometimes 11 win, sometimes 17, and the identity of the grand prize winner is wholly random. Over 52 weeks, a player buying one ticket a week has about a 7% chance of winning something at least once and about a 0.5% chance of hitting the grand prize at least once. Familiarity with these shapes keeps expectations honest.

The shared-jackpot problem

When many people play, several can pick the same winning set, and a jackpot is divided. The expected number of other winners rises with ticket sales. If 300 million tickets are sold for a game with 292 million combinations, the chance that a particular winning set was also picked by at least one other player is substantial, roughly 1 − e^(−300/292) ≈ 64% if numbers were chosen uniformly at random, and higher because people favour dates. This is part of why big jackpots are worth less than their headline, and why "avoid birthday numbers" is a sensible tip for sharing, not winning.

Probability words in plain language

• Independent events: one draw does not affect the next.
• Combination: a set of numbers where order does not matter.
• Expected value: the probability-weighted average return.
• Variance: how spread out real results are around the average.
• Law of large numbers: over very many plays, results settle near the average, but short runs can differ widely.
• Gambler's fallacy: believing a streak makes the opposite outcome due.

Near misses and "almost winning"

Matching four of five numbers feels like you were close. Mathematically you were no closer to the jackpot than someone who matched none: the draw is a fresh random sample and nothing about "close" improves your next chance. Casinos and lotteries know that near misses motivate people to play more. If you notice that near misses make you want to buy extra tickets, take it as a cue to stay within your budget.