The Birthday Paradox

Day 317 / 365

Today’s blog is about an interesting but counter-intuitive problem about probability. Suppose you are in a room with 22 other people, what are the chances that two people in the room have the same birthday?

If you had to guess, I am sure you would think it would be quite low, 5%, maybe 10. The answer is actually 50%. Yes, in a room with 23 people, the chances of 2 of them sharing a birthday would be 50–50!

What happens if we increase the number of people to 50. The probability goes up to 95%. At 75, the probability is about 99.95. So if you choose a random group of 75 people, it’s an almost certainty that two of them will have the same birthday. A class in school is on average 50 students. Did any two students in your class have the same birthday? I bet that was true.

So why does this happen? Suppose you toss a coin once. The probability of getting heads is 1/2 or 0.5. What if you toss it twice? probability of getting heads both the times is 0.5*0.5. Similarly, the probability of getting heads 10 times in a row is 0.5*0.5*0.5 …. 10 times, or 0.5 raised to the power 10.

Now let's look at the probability that two people will not have the same birthday. Since there are 365 days in a year, that probability will be 364/365

Now you can select two people out of 23 in (23*(23–1))/ 2 ways (everyone gets paired with everyone else other than themselves, and we divide by 2 as we would have counted each pair twice) i.e. 253.

Now if we want the probability that no two people in a group of 23 share the same birthday, that would be like tossing a coin 23 times and getting head. Thus that probability would be (364 / 365)²³, which comes out to be around 0.5.

As the number of people increases, the power also increases and the chances of not finding a match goes down.

This post is part of my 365 Day Project for 2019. Read about it here

Yesterday’s blog — How to Silence your inner critic