Home › Statistics & Probability › Lottery Odds Calculator

Lottery Odds Calculator

Computes the number of possible tickets, the jackpot probability and '1 in N' odds for a lottery where k numbers are drawn from a pool of n (optionally with a separate bonus-ball pool), plus a table of the odds of matching each number of main numbers.

When to use

You want the odds of winning or of matching a given number of balls in a lottery such as 6/49, Powerball (5/69 + 1/26) or EuroMillions (5/50 + 2/12).

Do not use when: Bonus balls are drawn from the same pool as the main numbers (UK Lotto 'bonus ball'), or you need the expected value of a ticket (use expected-value with the prize amounts).

Formula

P(match exactly m main numbers) = C(picks, m) × C(pool_size − picks, picks − m) / C(pool_size, picks); P(all bonus balls) = 1 / C(bonus_pool_size, bonus_picks); jackpot = 1 / (C(pool_size, picks) × C(bonus_pool_size, bonus_picks))

Hypergeometric probabilities for the main numbers, multiplied by the independent bonus-pool probability. Odds are for a single ticket per draw and ignore shared jackpots and prize amounts.

Inputs

ParameterTypeUnitRequiredDescription
pool_sizeintegeryesHow many different numbers the main balls are drawn from. Range: ≥ 2, ≤ 1000
picksintegeryesHow many main numbers are drawn (and chosen on a ticket) without replacement. Range: ≥ 1, ≤ 20
bonus_pool_sizeintegernoSize of a separate pool for bonus balls (e.g. 26 for the Powerball). Omit if there is none. Range: ≥ 1, ≤ 1000
bonus_picksintegerdefault 1How many bonus balls are drawn from the bonus pool (1 for Powerball, 2 for EuroMillions Lucky Stars). Range: ≥ 1, ≤ 5

Outputs

OutputTypeUnitDescription
total_combinationsnumberC(pool_size, picks) × C(bonus_pool_size, bonus_picks).
jackpot_probabilitynumber1 / total_combinations for one ticket.
jackpot_oddsstringOdds as '1 in N'.
matcheslistOne row per number of main matches (from picks down to 0): matched, ways (count as a string), probability and odds ('1 in N'), plus probability_with_bonus / odds_with_bonus and probability_without_bonus / odds_without_bonus when a bonus pool is given.

Example

6 of 49: {"pool_size":49,"picks":6} → {"total_combinations":13983816,"jackpot_probability":7.15e-8,"jackpot_odds":"1 in 13,983,816","matches":[{"matched":6,"ways":"1","odds":"1 in 13,983,816"},{"matched":5,"ways":"258","probability":0.0000184499,"odds":"1 in 54,201"},{"matched":4,"ways":"13,545","probability":0.000968619,"odds":"1 in 1,032"},{"matched":3,"ways":"246,820","probability":0.0176504,"odds":"1 in 56.66"},{"matched":2,"ways":"1,851,150","probability":0.132378},{"matched":1,"ways":"5,775,588","probability":0.413019},{"matched":0,"ways":"6,096,454","probability":0.435965}]}

Powerball: 5 of 69 plus 1 of 26: {"pool_size":69,"picks":5,"bonus_pool_size":26,"bonus_picks":1} → {"total_combinations":292201338,"jackpot_odds":"1 in 292,201,338","matches":[{"matched":5,"ways":"1","odds":"1 in 11,238,513","odds_with_bonus":"1 in 292,201,338","odds_without_bonus":"1 in 11,688,054"},{"matched":4,"ways":"320","odds_with_bonus":"1 in 913,129","odds_without_bonus":"1 in 36,525"},{},{},{},{}]}

GET https://tttkmbb.com/api/v1/calculate/lottery-odds?pool_size=49&picks=6

Machine access

Sources

FAQ

Does buying two tickets double my chances?

Yes, two different tickets give 2 / total_combinations, still about 1 in 7 million for a 6/49 game.

Why is 'match 5 without bonus' more likely than 'match 5 with bonus' by a factor of 25 in Powerball?

The bonus ball is an independent 1-in-26 draw: of every 26 tickets matching all five main numbers, on average one also matches the Powerball and 25 do not.

Related calculators