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April 22nd, 2013, 10:17 PM   #1
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simple probability

"Sam has 25 friends. What is the probability that at least one of Sam's friends has the same birthday as Sam?" Assume 365 days.

Okay, I see two ways to solve this, 1-p(no match) = 1-(364/365)^25 = .006287
or 25 x p(of a match) = 25(1/365) = .00685
obviously .006287 does not = .006845. I am pretty sure the first method is correct, but I really don't see where I am going wrong. Each friend has a 1/365 chance of having the same birthday as Sam,
and this chance happens 25 times ( 25 friends), thus a 25 times greater chance? I know I must be wrong, because if she had 366 friends, there would a probability greater than 1. I can also get the right answer if I think of it as a geometric series, but I really just can't shake off this innate, intuitive feeling that having 25 friends will increase the odds by 25 times.
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April 23rd, 2013, 09:37 AM   #2
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Your first method is correct, but 1-(364/365)^25 = 0.0662879... = 0.0663 approximately, not .006287.

Your second method is inaccurate, as it takes no account of the possibility that there is more than one match.
Note that 25(1/365) = 0.06849315... = 0.0685 approximately, not .00685.
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