The hypergeometric formula
Where:
- N — total pool size (e.g., 49 for a 6/49 lottery)
- D — numbers drawn by the lottery (often = M)
- M — numbers you pick on your ticket
- K — exactly how many of your picks match the draw
- C(n, k) — binomial coefficient ("n choose k")
Intuition: the numerator counts the number of ways to choose K winning numbers from the D drawn, then choose the remaining M−K losers from the N−D non-drawn. The denominator counts every possible ticket. Their ratio is the probability.
Worked examples
| Scenario | Inputs | P(K matches) | 1 in … |
|---|---|---|---|
| UK Lotto 6 of 49 — match 6 | N=49, D=6, M=6, K=6 | 0.0000072% | 13,983,816 |
| UK Lotto 6 of 49 — match 3 | N=49, D=6, M=6, K=3 | 1.77% | 57 |
| Powerball main 5 of 69 — match 5 | N=69, D=5, M=5, K=5 | 0.0000088% | 11,238,513 |
| Powerball main 5 of 69 — match 3 | N=69, D=5, M=5, K=3 | 0.180% | 556 |
Note: full Powerball jackpot odds also include the bonus ball. Use the jackpot probability calculator for game-level odds with bonus balls.
FAQ
Why "exactly" instead of "at least"?
"Exactly K" gives you the probability mass at one outcome. "At least K" sums all outcomes from K up to min(D, M). The full distribution table on this page gives you both — read across to "match exactly", or sum from K downward for "at least K".
Are these odds the same for any draw?
Yes — every lottery draw is independent, and the probability of any particular match count is identical for every draw. There's no "due number" or "hot number" advantage in a fair lottery; that's the gambler's fallacy.