What is the probablity that two people share the same month of their birthday?   

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If we assume that the likelihood of being born in any given month is the same as any other month then the answer is 1/12.

Consider rolling a pair of 12-sided die, with the numbers from 1-12 on the sides. There are 12 pairs (1,1),(2,2),...,(12,12) and there are 144 possible throws. Thus the probability of rolling doubles (which is the same as having matching birth months) is 12/144 or 1/12.

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A complication is that it is not equally likely to be born in any given month as some months have more days than others. If it is equally likely to be born on any day of the year, then the likelihood of being born in January , which has 31 days, is 31/365 while the probability of being born in February is 28/365.

We can compute the probability of a match by adding the probabilities that both are born in January to the probability that they were both born in February, plus the other months.

There are 7 months with 31 days, 4 months with 30 days and 1 month with 28 days (disregarding leap years.) So we have `7(31/365)(31/365)+4(30/365)(30/35)+(28/365)(28/365)~~.0834002627` Note that `1/12=.08bar(3)` so the results are close.

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A further complication is the assumption that you are equally likely to be born on any given day. There are slight increases in births in the months of March and April as well as August and September (9 months after June weddings/anniversaries and 9 months after Christmas and New Years -- at least in the Christian West.) Also the wrinkle of accounting for leap years (or lack thereof) makes the actual computation very complicated.

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