Lottery Odds Calculator

How it works

A lottery draw picks n numbers from a pool of N without replacement, and every set of n is equally likely. The number of possible draws is the binomial coefficient C(N, n), "N choose n": for a 6/49 game that is 13,983,816. A ticket wins the jackpot on exactly one of them, so the jackpot odds are 1 in 13,983,816.

For a smaller prize, count the draws that match exactly k of your numbers: choose which k of your n numbers are drawn, C(n, k), and which n − k of the other N − n numbers fill the rest of the draw, C(N − n, n − k). Dividing by C(N, n) gives the hypergeometric probability. "At least k" adds up the exact probabilities from k to n. The table lists every k so you can see where the chances go.

A bonus ball drawn from its own separate pool of M numbers is independent of the main draw, so matching it multiplies the odds by M. Powerball draws 5 from 69 plus one from 26, so its jackpot odds are C(69, 5) × 26 = 1 in 292,201,338.

Formula

C(N, n)              = N! / (n! (N − n)!)
P(exactly k)         = C(n, k) C(N − n, n − k) / C(N, n)
P(at least k)        = Σ P(exactly j)   for j = k … n
with bonus ball      = P(main) / M
odds                 = 1 in 1 / P

Example

In a 6/49 lottery there are C(49, 6) = 13,983,816 possible draws, so the odds of matching all six are 1 in 13,983,816 (0.0000007151%). Matching exactly five is C(6, 5) × C(43, 1) = 258 ways, or 1 in 54,200.84, and matching at least three is 1 in 53.66.

Powerball draws 5 numbers from 69 and one Powerball from 26. The jackpot odds are C(69, 5) × 26 = 11,238,513 × 26, or 1 in 292,201,338.

Assumptions and limitations

• Every combination is equally likely: a fair draw without replacement, and no bias in the numbers.
• The ticket holds the same count of numbers as the draw (n), which is how most pick-n lotteries work. Games where you pick more or fewer numbers than are drawn (some Keno bets) are not modelled.
• The bonus ball is drawn from its own separate pool, independently of the main numbers (Powerball, Mega Millions). A bonus ball drawn from the remaining main-pool numbers (UK Lotto's bonus ball) has different odds and is not modelled.
• Odds are per ticket. Buying more tickets with distinct number sets multiplies the chance of winning accordingly, but shared jackpots and prize values are not considered.
• Combination counts above about 9 quadrillion (2^53) are computed in floating point and may be off in the last digit; the odds are still correct to many significant figures.

Frequently asked questions