How many people have to be in a room in order that the probability that at least two of them celebrate their birthday in the same month is at least 1/2? Assume that all possible monthly outcomes are equally likely.

Question

idealist10 · Answer

@ganeshie8 @iambatman @Zarkon

zarkon · Answer

this is a variant of the birthday problem http://en.wikipedia.org/wiki/Birthday_problem use 12 instead of 365

idealist10 · Answer

@amistre64 @sangya21

anonymous · Answer

The probability that there are at least 2 people in a group of n who have birthdays in the same month is just 1 minus the probability that all n have birthdays in distinct months. n people can fit distinctly into 12 slots in

$$12\times 11 \times \cdots \times (12 - n + 1)$$
different ways, and the size of the sample space – the number of ways to put 12 people in 12 slots with overlap allowed – is just 12^n, yielding the probability of a match being

$$P(\text{at least two birthdays in the same month})
= 1 - \frac{12\times 11 \times \cdots \times (12 - n + 1)}{12^5}$$

Finding n by trying various n shows that when n=5, the probability of a match is 89/144, or about 0.618056. When n=4, this probability is only .427083, so n=5 is the answer

anonymous · Answer

http://www.stat.washington.edu/~hoytak/teaching/current/stat394/