Lottery Odds Calculator

A lottery draws k numbers from a pool of N. The order in which they come out does not matter, so the number of different draws is the number of combinations C(N, k) = N! / (k! × (N − k)!). Every draw is equally likely, so a ticket with k numbers wins the jackpot with chance 1 / C(N, k). In a lottery that draws 6 numbers from 49 that is 1 in 13,983,816.

Many games add a bonus draw from a second pool, such as the red Powerball or the Lucky Stars of EuroMillions. The two draws are independent, so the number of different tickets is the product of the two counts and the jackpot chance is 1 divided by that product.

• Jackpot odds. Enter the size of the main pool and how many numbers are drawn from it. If the game has a bonus draw, also enter the size of the bonus pool and how many bonus numbers are drawn; otherwise leave those two fields empty.
• Odds of matching some numbers. Also enter how many of the main numbers, and how many of the bonus numbers, a ticket matches. The result is the chance of matching exactly that many, which is a different prize from the jackpot.
• Chance with several tickets. Enter the number of different tickets. Two different tickets cannot both match the whole draw, so their chances simply add up.
• A ticket carries as many numbers as are drawn. The pools can hold up to 1,000 numbers, and the number of tickets can have up to 30 digits.
• The bonus numbers come from their own pool. A bonus ball that is drawn from the main pool, as in the UK Lotto, is a different model, which is worked out by hand below.

Here N is the size of the main pool, k the numbers drawn from it, M the size of the bonus pool and b the bonus numbers drawn. For a game without a bonus draw, drop the bonus factors. To match exactly m of the k drawn numbers, a ticket takes m of the k winning numbers and the other k − m from the N − k numbers that were not drawn; that is the hypergeometric count.

Powerball draws 5 white balls from 69 and one red Powerball from 26. Choose Jackpot odds and Load example to enter 69, 5, 26 and 1.

• White balls: C(69, 5) = 69! / (5! × 64!) = 11,238,513 ways.
• Powerball: C(26, 1) = 26 ways.
• Every white set pairs with every Powerball: 11,238,513 × 26 = 292,201,338 equally likely tickets.
• Exactly one ticket matches the draw, so the chance of the jackpot is 1 / 292,201,338, which is 3.4223e-7%, or 1 in 292,201,338. This is the jackpot figure Powerball publishes.

Choose Odds of matching some numbers and Load example, which enters the Powerball game with 5 main numbers matched and 0 bonus numbers matched.

• White balls: all 5 of the 5 drawn, C(5, 5) = 1 way, and none of the other 64, C(64, 0) = 1 way: 1 × 1 = 1 way.
• Powerball: none of the 1 drawn, C(1, 0) = 1 way, and the other 1 from the 25 numbers that were not drawn, C(25, 1) = 25 ways: 1 × 25 = 25 ways.
• Winning tickets: 1 × 25 = 25 of the 292,201,338, so the chance is 25 / 292,201,338, which is 8.5557e-6%, or 1 in 11,688,053.52.

These agree with the odds that Powerball publishes for each prize level. Adding the nine winning counts gives 11,750,538 of the 292,201,338 tickets that win something, so the chance of some prize is 1 in 24.87. To reproduce a row, choose Odds of matching some numbers and enter the two match counts.

The Mega Millions row uses the format in force since April 2025. Operators change their formats from time to time, so check the current rules of your game and enter its pool sizes.

Every ticket matches exactly one of these counts, so the seven rows add up to all 13,983,816 tickets. For three matches, C(6, 3) = 20 ways to take three of the six drawn numbers times C(43, 3) = 12,341 ways to take the other three from the 43 numbers that were not drawn gives 20 × 12,341 = 246,820 tickets.

Some games, such as the UK Lotto and Lotto 6/49, draw an extra bonus ball from the numbers left in the main pool, and a ticket that matches five numbers plus that ball wins a bigger prize. The calculator does not model this, but the count is short: a ticket wins it when it holds five of the six drawn numbers and the bonus ball as its sixth number, which can happen in C(6, 5) = 6 ways, one for each drawn number it leaves out.

For a 6 of 59 lottery the chance is 6 / 45,057,474 = 1 in 7,509,579, and for a 6 of 49 lottery it is 6 / 13,983,816 = 1 in 2,330,636. These 6 tickets are part of the 258 that match exactly five numbers in the 6 of 49 table, which leaves 252 for the five-number prize without the bonus ball.

The chance grows in proportion to the number of tickets, because different tickets cannot win together. Even 10,000 tickets leave the chance below four thousandths of one percent. To cover every combination of the game, you would need 292,201,338 tickets.

Each draw is independent, so a ticket for the next draw has the same 1 in 292,201,338 chance as the last one. The chance of winning at least once in n draws is 1 − (1 − 1/292,201,338)^n, and it builds up slowly. Powerball has three draws a week, which is 156 a year.

A 50% chance needs 202,538,534 draws, which is about 1.3 million years at three draws a week.

No. Every combination is equally likely, including 1-2-3-4-5-6 and a quick pick chosen by the machine. Past draws do not influence later ones, so "hot" and "cold" numbers change nothing, and believing that a number is "due" is the gambler's fallacy. What your choice can change is how many other players hold the same numbers, and therefore how many people share a jackpot if it is won; it never changes the chance of winning it.

• Fair, independent draws. Every number is equally likely and no draw depends on an earlier one. The calculation cannot be better than that assumption.
• One ticket and one draw. The odds are for a single ticket in a single draw. The chance of winning at least once over many draws is larger, as the table above shows, but still tiny for a jackpot.
• Different numbers on a ticket, order ignored. Digit games, such as three digits from 0 to 9 in the exact order, are not of this kind. They have 10 × 10 × 10 = 1,000 outcomes.
• Odds of winning are not the prize you receive. Jackpots can be shared by several winners and are taxed in many places. To judge a ticket, multiply each prize by its chance and add the results, then compare the total with the price.
• Formats change. Pool sizes and prize levels are set by the operator and have been changed before, so enter the numbers of the game as it is played today.

The counts are the same as the calculator's. The hypergeometric commands give the chance for the white balls alone, before the Powerball is taken into account, so they are the main-number part of a prize.