 September 10, 2019

The Birthday Paradox is not a “paradox” in the true sense of the word. It’s more of an “astonishing realization” once people are presented with the proof.

It asks: How many people are needed in a room before there is a 50% chance that two people share a birthday? The answer, surprisingly, is only 23.

How can this be?*

Logically, we assume that this means one individual is asking 23 people if they share the same birthday.

This individual has a one in 365 chance that they share a birthday with one of the others, which is a 0.27% chance. If I ask 23 people, then it’s 23*0.27 = 6.21%. The odds are still pretty low.

With the Birthday Paradox, what we’re really looking at is each person asking each other person if they share a birthday. The odds change dramatically when you put 20+ people in a room and they each ask everyone else if they share a birthday.

We know, a priori, that the magic number to have a 50% chance is 23 people in the room.

Let’s do some math…

Remember, a probability of “1” means a 100% chance of the event happening. So,

“the probability of two people sharing the same birthday” = 1 – (the probability that no one shares a birthday)

In a list of 23 people, comparing the birthday of the first person on the list to the others allows 22 chances for a matching birthday, but comparing every person to all of the others requires a little calculation.

How many pairs can 23 people make? Since any one person out of 23 can make 22 different pairs, multiply 22 by the total number of people (23) and divide by the amount of people it takes to make a pair (two). This equals 253.

23*22/2 = 253

Now, you have to ask the question “how likely is it for a pair of people to not share a birthday in a year?” A person’s birthday is 1/365 of the year. We can assume someone else can be born on any of the 364 days of a 365 day year, and not share the same birthday.

This can be written as as:

1-(1/365) = 364/365 = 0.9972

For all 253 pairs to be different, we need to multiply those odds (for example, this is like saying: “The number of possible outcomes if three dice are thrown is 6 x 6 x 6 = 216″:

(.9972)^253 = 0.4995

The chance of a birthday match out of 23 people is then:

1 – 49.95% = 50.05%.

*For these calculations, I am assuming:

1. There are 365 days in a year (ie, not a leap year)
2. There are no twins in the group of 23

I highly recommend the tutorial at Better Explained to brush up on the mathematics behind permutations and combinations – I did!

Observe the graph below – note that to have a 90% chance, you need just over 40 people in the room! It’s amazing how easily we can be fooled – something that seems nearly impossible, upon further investigation, is suddenly highly probable. But how is the Birthday Paradox useful in our Skeptical Toolkit? It helps us to digest the actual odds of “coincidences”.

Coincidences” will be reviewed in a future post… meanwhile, check out this video: