Lottery Odds Calculator
Odds of winning a 'pick N of M' lottery, with optional bonus ball.
Overview
The Lottery Odds Calculator computes the probability of winning a "pick N of M" lottery — the most common format used by national and state lotteries worldwide. Tell it the number of balls drawn, the size of the number pool, and whether the lottery includes a separate bonus ball, and it returns the exact odds for the jackpot plus secondary prize tiers.
The numbers are usually staggering: typical national lotteries advertise jackpot odds around 1 in 100 million to 1 in 300 million. The calculator helps put those numbers in perspective by also showing the expected number of tickets you'd need to buy for an even chance of winning, and the implied ticket value at various jackpot sizes.
How it works
A pick-N-of-M lottery has C(M, N) equally likely outcomes, where C is the binomial coefficient. The probability of matching all N drawn balls with a single ticket is 1 / C(M, N). For tiers that match fewer balls, the count of qualifying outcomes is C(N, k) * C(M - N, N - k), giving the probability C(N, k) * C(M - N, N - k) / C(M, N) of matching exactly k of N.
A bonus-ball mechanism — common in Powerball, Mega Millions, EuroMillions — draws a separate number from a separate pool. The jackpot then requires matching all N regular numbers plus the bonus, multiplying the odds by the bonus-pool size. Some tiers reward matching some regular numbers plus the bonus, which slightly increases the chance of any prize but rarely changes the jackpot odds materially.
Examples
- Pick 6 from 49 (UK Lotto-style): jackpot odds 1 in C(49,6) = 13 983 816.
- Pick 5 from 70 + 1 of 25 (Mega Millions-style): jackpot odds 1 in C(70,5) * 25 = 302 575 350.
- Pick 5 from 69 + 1 of 26 (Powerball-style): jackpot odds 1 in C(69,5) * 26 = 292 201 338.
- Pick 5 from 50 + 2 of 12 (EuroMillions-style): jackpot odds 1 in C(50,5) * C(12,2) = 139 838 160.
FAQ
Are my odds better if I buy multiple tickets?
Yes, linearly — two tickets double the odds. Buying every possible combination guarantees a jackpot but costs more than the jackpot at most prize levels.
Is there a smart number to pick?
No — every combination has the same chance. Avoiding popular sequences (1-2-3-4-5-6, birthdays) only matters if you win, because it reduces the chance of sharing the jackpot.
Why does the expected value of a ticket seem negative?
Operators take a cut, taxes apply, and large jackpots get split among multiple winners. Expected value almost always favours not playing.
Do hot or cold numbers matter?
No. Each draw is independent and uniform. The "hot number" myth is a sampling-variance illusion.
What's the worst-odds major lottery?
Italy's SuperEnaLotto jackpot is roughly 1 in 622 million — among the longest odds of any regulated lottery.