What is the probability that nobody is born in the same month?

Question

You have 12 people in a room, what is the probability that nobody is born in the same month?

So far i have: $\frac{12!}{12^{12}}$ but i am not sure if this is right.

If anyone could confirm this is the way to go or tell me where i am wrong it would help me alot.

Thank you.

Your solution is correct. Preassumed of course that the months have equal chances to be elected as month of birth. — drhab, Oct 11 '15 at 21:13
Yeah, more complicated if you take into account the probabilities of each month is different. — Thomas Andrews, Oct 11 '15 at 21:19

score 2 · Answer 1 · answered Oct 11 '15 at 21:38

You can model this as ball and urn problem and say something like this:

You have 12 balls and 12 urns. Randomly pick ball and put it into random urn, what is probability that, at the end, every urn have exactly one ball.

Solution: There is 12! ways to arrange balls into urns so that every urn have exactly one ball. If you at random pick ball and put it into random urn there is $12^{12}$ ways to do that. So at the end you solution is good $\frac{12!}{12^{12}}$.

score 1 · Accepted Answer · answered Oct 11 '15 at 21:21

1

If $p_i$ is the probability of being born in month $i$, then then the probability that twelve random people are born in twelve months is:

$$12!p_1p_2\dots p_{12}$$

If you are assuming $p_i=\frac{1}{12}$ for each $i$, then your answer is correct.

answered Oct 11 '15 at 21:21

Thomas Andrews

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What is the probability that nobody is born in the same month?

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