Birthday Paradox

Let's consider there are 365 days in a year (sometime it has 366 but we will not consider that case). We want to find out how big a group needs to be such that there is a 50% chance that two people in this group have the same birthday.

You may think that well it needs about half of 365 people (which is about 183) people in the group. Well the actually is acutally much smaller than that.

To analyze this problem, let's start with a smaller group of people, for example 3 people. The probability that in the group of 3 people, at least two people has the same birthday is:

\[ 1-\frac{{}_{365} {P}_{3}}{365^3} \approx 0.008204 \]

\(\frac{{}_{365} {P}_{3}}{365^3}\) is the probability that all 3 people in the group has different birthday. 1 minus that gives the probability that at least two people in that group share the same birthday.

Using this method, we can find the probability that at least two people has the same birthday in a n people group (where \(2\leq n \leq 365\)) is:

\[ 1-\frac{{}_{365} {P}_{n}}{365^n} \]

The table below show the number of people in a group and the probability that at least two people at the corresponding group has the same birthday: