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MasterMath

Lottery Odds Calculator

How many combinations a lottery has and what the real chance is of matching them all. It works for any drum-based draw, with or without a bonus ball.

Probability

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Probability—
As a percentage—
Combinations in the main draw—
Combinations in the bonus draw—
Total combinations—
Expected value of a ticket—
Cost of buying every line—

How this was worked out

    Indicative result. The expected value ignores the smaller prizes and the possibility of splitting the jackpot with other winners. In practice, no lottery pays out more than it takes in.

    The formula

    combinations = C(n, k) = n! ÷ (k! × (n − k)!)

    Where it comes from

    Because the order the balls come out in makes no difference, what counts is combinations rather than permutations. With 49 numbers and 6 to match there are 13,983,816 distinct combinations, and only one wins. If there is a second drum, the combinations of the two multiply together.

    How to work it out by hand

    1. Work out the combinations in the main drum with the C(n, k) formula
    2. Do the same for the bonus drum if there is one
    3. Multiply the two figures together
    4. The probability is one in that total

    What is worth knowing

    To put the number in perspective: matching a 6/49 draw is roughly as likely as calling a coin toss correctly twenty-three times in a row. And there is a detail that often gets missed: even when the jackpot exceeds the cost of buying every combination, the expected value stays poor, because if you win it is likely someone else did too and the jackpot splits. Picking unpopular numbers does not improve your chance of winning, but it does improve your chance of not sharing.

    Frequently asked questions

    What are the odds of winning the lottery?

    On a 6/49 draw, one in 13,983,816. That is 0.0000072 %.

    Why combinations rather than permutations?

    Because the order the balls come out in does not change the winning ticket.

    Does picking certain numbers help?

    Not for winning: every combination is equally likely. It does help avoid sharing the jackpot, if you pick unpopular numbers.

    Can buying every combination pay off?

    Almost never. On top of the cost, if the jackpot is shared with other winners the whole thing runs at a loss.