How lottery odds are calculated, with Powerball worked out by hand

Every odds figure printed on a lottery ticket comes from one piece of counting. Here is that counting, done in full for all nine Powerball prize tiers.

Lottery · Updated 5 October 2026 · 4 minute read

The odds of a lottery are not an estimate, and nobody measures them by watching draws. They are counted. A game publishes its rules (how many balls are in the drum, how many are drawn) and the odds of every prize follow from those rules by arithmetic that fits on one page. This guide does that arithmetic for Powerball, and the same method works for any draw game.

The rules that set the odds

A Powerball draw takes 5 white balls from a drum of 69 and 1 red ball from a separate drum of 26. The order the white balls come out in does not matter. Lottery people call this pair of numbers the game's matrix, and it is the only input the odds need.

Step one: count the possible draws

The number of ways to choose 5 balls from 69, ignoring order, is written C(69, 5). You can work it out by multiplying down from 69 five times and dividing by the number of ways to arrange five things:

(69 × 68 × 67 × 66 × 65) ÷ (5 × 4 × 3 × 2 × 1) = 11,238,513

Each of those sets of white balls can be paired with any of the 26 red balls, so the number of different results a draw can produce is:

11,238,513 × 26 = 292,201,338

Your ticket names exactly one of them. That is where the jackpot odds of 1 in 292,201,338 come from, and it is the whole derivation.

Step two: count the ways to win each smaller prize

For the lower tiers, split the 69 white balls into the 5 that were drawn and the 64 that were not. To match exactly three white balls, your ticket must hold 3 of the drawn five and 2 of the undrawn sixty-four:

C(5, 3) × C(64, 2) = 10 × 2,016 = 20,160 ways

The red ball then either matches (1 way) or does not (25 ways). Divide the total number of draws by the number of winning ways and you have the odds for that tier.

Add the winning ways together and you get 11,750,538. Divide the total by that and the overall odds of winning any prize come to 1 in 24.87, the figure printed on the play slip.

Two things in that table that surprise people

Matching only the Powerball is 1 in 38, not 1 in 26. There are 26 red balls, so it feels as though the last row should be 1 in 26. The prize, though, is for matching the red ball and none of the white ones. Some of the tickets that match the red ball also match one or more white balls and are paid in a higher tier, which removes them from this row.

Three white balls is easier than two white balls plus the Powerball. Needing the red ball multiplies the difficulty by 26, which outweighs the saving from needing one fewer white ball. That is why both tiers pay the same $7.

What the odds cannot tell you

Odds describe one ticket in one draw. They say nothing about which numbers will come up, because every one of the 292,201,338 combinations is exactly as likely as every other, including 1-2-3-4-5 with Powerball 6. They also do not change with the size of the jackpot or the number of people playing. More players means a greater chance that somebody wins and a greater chance that the jackpot is shared, and your own ticket's chance stays where the matrix put it.

Buying more tickets does improve your chance in a strictly linear way: ten different tickets are ten chances in 292 million. Ten times a very small number is still a very small number.

Checking any other game

The method does not change. Find the matrix in the game's official rules, compute the number of possible draws, then for each tier count the tickets that would match exactly that many drawn and undrawn balls. When a lottery changes its matrix, every figure changes with it, which is why odds quoted in an old article can disagree with the ones on today's ticket. If a published figure does not agree with the arithmetic, the arithmetic is the one to trust.