You may solve the simple probability, *“Given a set of n randomly chosen people, what is the probability of two people having same birthday?”*

You may also solve, * “At least how many people need to be picked to make the probability 100% that at least two of them have same birthday?”*

Answer to the second question is 367, because there are 366 possible birthdays (including Feb-26).

The question now is, *“At least how many people need to be picked to make the probability 99.9% that at least two of them have same birthday?”*

Answer is 70.

And if the question is,*“At least how many people need to be picked to make the probability 50% that at least two of them have same birthday?”*

Answer is 23.

Let us first the first question,*“Given a set of n randomly chosen people, what is the probability of two people having same birthday?”*

If P is the probability then P’ is the probability of no two people (out of n) have the same birthday.

P = 1 – P’

P’ can be calculated as below

For 23 people, this comes out to be

P’ = 1 * (364/365) * ( 363/365)*… * (343/365)

Solving the equation, we get

P’ = 0.492703

P = 1-P’ = 0.5 (approx)

For different values of n, following are the probabilities:

(Refer: https://en.wikipedia.org/wiki/Birthday_problem)

Please refer the above link to see how to derive n from probability.