About

Lotteries are often called a tax on people who are bad at math, but understanding the underlying probability theory transforms them from a blind gamble into a study of combinatorics. Whether you are analyzing the massive pools of the US Powerball or a local state raffle, the math remains the same: it is a problem of calculating combinations without replacement.

This tool determines the exact probability of matching a specific set of numbers from a larger pool. Unlike simple percentage calculators, this engine accounts for standard "Main Ball" pools as well as distinct "Bonus Ball" pools (like the Powerball or Mega Ball), providing the true "1 in X" odds. Understanding these astronomical numbers helps in managing expectations and underscores why lottery syndicates (pooling tickets) are a popular strategy to marginally improve these odds.

Formulas

The core of lottery mathematics is the Combination formula, which calculates how many unique groups of size k can be chosen from a pool of n items.

C(n, k) = n!k!(n − k)!

For lotteries with a Bonus Ball (like Powerball), the total combinations are the product of the main pool combinations and the bonus pool combinations:

Total Odds = C(n₁, k₁) × C(n₂, k₂)

Reference Data

Lottery Name	Main Numbers	Pick	Bonus Numbers	Pick	Jackpot Odds (1 in...)
US Powerball	1 - 69	5	1 - 26	1	292,201,338
US Mega Millions	1 - 70	5	1 - 25	1	302,575,350
EuroMillions	1 - 50	5	1 - 12	2	139,838,160
EuroJackpot	1 - 50	5	1 - 12	2	139,838,160
UK Lotto	1 - 59	6	0	0	45,057,474
SuperEnalotto (Italy)	1 - 90	6	0	0	622,614,630
La Primitiva (Spain)	1 - 49	6	1 - 9	1	139,838,160
Oz Lotto (Australia)	1 - 45	7	0	0	45,379,620
Lotto 6/49 (Canada)	1 - 49	6	0	0	13,983,816
Irish Lotto	1 - 47	6	0	0	10,737,573

Frequently Asked Questions

This is due to the nature of factorials in combinatorics. Increasing the pool size (n) even by 1 increases the numerator in the factorial equation significantly. For example, moving from a 'Pick 6 out of 49' to 'Pick 6 out of 50' adds millions of new potential combinations, exponentially decreasing your chances of covering them all.

Mathematically, no. Lottery draws are independent events. The probability of your specific set of numbers being drawn is exactly the same in every single draw, regardless of past history. However, playing consistently ensures you don't 'miss' a draw if your numbers theoretically come up.

While often used interchangeably, they are distinct. Probability is the ratio of favorable outcomes to total outcomes (e.g., 1 chance in 292 million). Odds compare the favorable outcomes to the unfavorable outcomes (e.g., 1 to 292 million minus 1). In lotteries with such massive numbers, the difference is negligible, but statistically, they are different measures.

Bonus balls usually come from a separate machine or pool. This requires a 'multi-stage' calculation. You must first calculate the probability of matching the main balls, and then multiply that by the probability of matching the bonus ball(s). This multiplication is why lotteries like Powerball have such high difficulty levels.