Question

how many people do you need to gather to be sure that at least two of them have birthdays in the same month?

Answer 1

You would need 13. Since there are 12 months, if you gather 12 people and they each have different months, there would be 1 person per month. If you add 1 more person, it is safe to assume that that person will be with another person in a certain month.

Answer 2

When 23 people are gathered, there is more chance than not that 2 of them have the same birthday. In order to help understand this, you should consider how many PAIRS of people there are. With 23 people there are 253 different pairings i.e. more than half of the 365 possible birthdays.