So I'm working on a probability problem:
In Exercise 19 assume it is equally likely that a person is born in any given month of the year.
- b) What is the probability that in a group of $n$ people chosen at random, there are at least two born in the same month of the year?
I know that the probability of two people being born in the same month is $1/12$ (from part a of this problem) and that the pigeonhole principle shows that some $n≥12$ will guarantee that two people within the group were born within the same month.
I don't know what to do with this information though, especially when looking at the book answer: $1 − 11/12 · 10/12 ··· (13−n)/12$. It doesn't make sense to me that a group ($n$) containing only 1 person with this formula would have a 100% chance of being born in the same month as someone else in the group. You need two people to compare birth months, so how can this be right? I thought that after 12 people, you'd be guaranteed to find two people with matching birth months. If I were to make a formula, it would look more like: $(n-1)·(1/12)$