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Find the probability of at least one birthday match among a group of 48 people.

  • math -

    The probability is

    1 - Probability of no matches

    The probability of no matches is the ratio of the number of ways you can choose the birthdays such that each person has a different birthday (let's call this N) divided by the number of ways to distribute birthdays without restriction (let's call this M).

    CLearly M = 365^48


    N = 365!/(365 - 48)!

    To see this, consider that the first person can have 365 birhtdays, the second can have 365-1 as his/her birthday bust be different from the first, The third can have 365 - 2 birthdays, etc. etc.

    Then, to evaluate the numbers you have to be careful if your calculator does not display numbers larger than 10^100. You can then take logarithms and use Strirling's formula to evaluate the factorials, or simply sum the 48 terms in the expression for Log(N).

    Yo should fiund that 1 - N/M = 0.9606

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