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Expert: - 12/28/2010


Question
Hello,

Could you please help me with the following question?

What is the probability that out of 23 randomly chosen people, at least two share a birthday? Assume that all 365 days of the year are equally likely and ignore leap years. (Hint:find the probabilities that two people have different birthdays, that three people have different birthdays, so on.) (Ans: 0.5073)

Answer
I know a way to do it that is much easier than trying to follow the hint.
Calculate the probability that all of them are different first.
The answer is then one minus this amount.

To find the chance that they are all different, the first person could have 365 days to have a birthday on.  The 2nd person could have 364 choices left.  The 3rd person would have 363 choices left.  The 4th person woyld have 362 choices left.

To comute the answer, divide each one by the total days and multiply together.
That would be (365/365)(364/365)(363/365)...(344/365)(343/365).
That is the probability they each have separate birthdays.
Call this value p.

To compute that probability that at least two share a birthday, it would be 1 - p.

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