What is the probability that 4 randomly selected people all have different birthdays?

Let us assume that there are exactly 365 days in a year and that birthdays are uniformly randomly distributed across those days.

First, what is the probability that 2 randomly selected people have different birthdays? The second person's birthday can be any day except the first person's, so the probability is 364/365.

What is the probability that 3 people will all have different birthdays? We already know that there is a 364/365 chance that the first two will have different birthdays. The third person must have a birthday that is different from the first two: the probability of this happening is 363/365. We need to multiply the probabilities since the events are independent; the answer for 3 people is thus 364/365 × 363/365.

You should now be able to solve it for 4 people.