The Birthday Problem

Using the naive definition of probability

If we have k people then we want to find the probability that at-least 2 have the same birthday.

Before we solve this problem we are going to make some assumptions :

• We are going to exclude Feb 29th
• The other 365 days are equally likely
• Independence of birth (i.e. we ignore twins)

If k is 365 then the probability is 1.
Now if k is less then 365 then,

• Let us find the probability of finding no match in birthdays. So as for the denominator each person has a choice of 365 days, if we have k people, then in total we have $365^k$ choices. Now since we are calculating the probability of no match, if the first person has 365 choices, then the second person must have 364 choices and so on till the $k^{th}$ person has 365 - k + 1 choices.

$$\text{P(no match)}=\frac{\mathrm{365.364.363......}(365-k+1)}{365^k}$$

• For finding the probability of matching birthday, we must subtract the above probability from 1

$$\text{P(match)}=1-\text{P(no match)}=1-\frac{\mathrm{365.364.363......}(365-k+1)}{365^k}$$

By mathematical approximation we get :

$$\text{P(match)}\approx 1-e^{-k(k-1)/730}$$

If we plug in k = 23 in this formula we get a probability of $50\%$.
If k = 50 then the probability is $96.513\%$ and if k = 100 then it is $99.999\%$.

This result is counter-intuitive. To get some intuition, think of it as choosing a pair from k people that is $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ choices. So if we have 23 people there can be 253 pairs and now it seems to be possible that at least one of this pair has the same birthday.