Lottery odds calculator
Math & numbersHow likely is it to win the lottery?
The number of possible rows in a lottery draw and the chance that one row hits the jackpot.
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Jackpot chance per row
1 in 18,643,560
A draw of 7 numbers out of 40 has 18,643,560 possible rows, and only one of them hits the jackpot.
More about the count
- Formula
- C(40, 7) = 40! ÷ (7! × 33!)
- Possible rows
- 18,643,560
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The number of possible rows in a lottery draw and the chance that one row hits the jackpot.
How to use
- Enter how many numbers are in the draw and how many you pick on one row, such as 7 out of 40.
- Read the chance that one row hits the jackpot, with the number of possible rows.
- For a game with two drums, count each drum on its own and multiply the rows.
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How it works
A lottery row wins the jackpot only if it matches every drawn number. The order of the draw does not matter and a ball never comes back, so the number of possible rows is the number of combinations: C(n, k) = n! ÷ (k! × (n − k)!), with n the numbers in the draw and k the numbers on a row.
Drawing 7 numbers out of 40 gives 40! ÷ (7! × 33!) = 18,643,560 rows, so one row has a 1 in 18,643,560 chance. Ten rows have ten times the chance, still 1 in 1,864,356.
The odds hold for a fair draw, where every row is equally likely. Smaller prizes for fewer correct numbers have their own, better odds, and a shared jackpot is split between the winners.
Worked example
How many pairs can you choose from 5 distinct items without repeats?
- There are 20 ordered picks, but each pair appears in 2 orders.
Ignoring order gives 20 ÷ 2 = 10 pairs.
Frequently asked questions
What are the odds of a lottery that draws 7 numbers out of 40?
There are C(40, 7) = 18,643,560 possible rows, so one row has a 1 in 18,643,560 chance of hitting all seven.
How do I work out the odds of a game with two drums?
Count each drum on its own and multiply. Picking 5 of 50 and then 2 of 12 gives 2,118,760 × 66 = 139,838,160 possible rows.
Do some numbers win more often?
No. In a fair draw every row is equally likely, so no pattern or past result changes the odds.
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