How lottery odds are calculated.

Counting the ways

A lottery draw picks a handful of numbers from a pool, and the order does not matter. The question is how many different sets of that size can be picked from the pool. Mathematicians call this a combination and write it as C(n, k), the number of ways to choose k items from n. The formula is n! divided by k! times (n minus k)!, where the exclamation mark means factorial: multiply the number by every whole number below it down to one.

For a game that picks 6 numbers from 49, that is 49! over 6! times 43!, which works out to 13,983,816. Every one of those sets is exactly as likely as every other, so a single line has a one in 13,983,816 chance of matching all six.

A shortcut you can do by hand

Factorials of 49 are not fun, but the formula simplifies. For six from 49 it becomes 49 × 48 × 47 × 46 × 45 × 44 on top and 6 × 5 × 4 × 3 × 2 × 1 underneath. The top is the number of ways to draw six balls in order, and the bottom removes the order, because the ticket does not care which ball came out first.

Where the bonus ball goes

Games like Powerball and EuroJackpot draw from two separate pools. Powerball takes five numbers from 69 and then one more from a second pool of 26. The two draws are independent, so their odds multiply. Five from 69 is 11,238,513 combinations. Multiply by 26 for the Powerball and you get 292,201,338. That single extra ball makes the jackpot 26 times harder to hit, which is exactly why it exists: it lets the operator advertise a bigger prize without paying it out more often.

EuroJackpot and EuroMillions draw two bonus numbers from 12. Two from 12 is 66 combinations, and five from 50 is 2,118,760, so the jackpot odds are 66 times 2,118,760, or one in 139,838,160.

Why smaller prizes exist

Matching some of the numbers is far more likely than matching all of them, and the same counting method gives the figure. To match exactly three of six in a 6-from-49 game, count the ways to pick three of the six winning numbers, C(6, 3) = 20, times the ways to pick three of the 43 losing numbers, C(43, 3) = 12,341. That is 246,820 favourable sets out of 13,983,816, or about one in 57. Operators pay small fixed prizes for those matches because they happen constantly, and it is those frequent small wins that keep people buying.

What a ticket is worth

Expected value is what a ticket would earn on average if you could play the same draw forever. Multiply each prize by its probability, add them up, and compare to the ticket price. For nearly every draw the answer is well under half the price, because the operator keeps a large share for prizes in other tiers, costs, tax and profit. A ticket is not an investment, and no choice of numbers changes that.

The one thing you can influence is how many people you would share a jackpot with. Numbers 1 to 31 are heavily over-played because of birthdays, and rows, diagonals and multiples of seven get picked by many people at once. A random line is no more likely to win, but if it does it is less likely to be split. That is the honest case for a random line, and it is the only one.

Randomness with no memory

Each draw is independent. A number that has not appeared for a year is not due, and a number that came up last week is not tired. The balls do not know the history, so strategies built on hot or cold numbers are counting patterns in noise. The Lottery Number Generator draws every line fresh from your browser's cryptographic random source, with the ranges for each game built in and a custom mode for anything else.

Lotteries are entertainment with a known cost. Decide the cost in advance, and if it stops being entertainment, operators offer limits and self-exclusion, and most countries have a helpline.