Birthday Paradox – Canadian Association for Girls In Science (CAGIS)

Birthday Paradox

This year it’s our 30th birthday, so what better topic to talk about than the birthday paradox!

With 23 people in a room, there is a 50% chance that at least two people will have the same birthday. With 75 people in a room, there is a 99% chance that at least two people will have the same birthday.

Before we explain the birthday problem, there are a few assumptions we need to make about the people in the room:
● There are no twins
● There are 365 days in a year
● People are equally likely to be born on any day of the year

Number of Comparisons
The first person’s birthdate will be compared to the birthdates of the remaining 22 people in the room. The second person’s birthday will be compared to the remaining 21 people in the room because their birthdate has already been compared to the first person, leaving 21 people remaining. The third person’s birthdate will be compared to the 20 remaining people in the room because their birthday has already been compared to the first two people.

This comparison continues, and the total number of comparisons in the room will be 22 + 21 + 20 … + 0 = 253 total comparisons! That is a lot of comparisons, and it is why we are likely to find a match!

Percent chance of a match
Probability is the likelihood that something will happen. In the case of the birthday paradox, it is the likelihood that at least two people in a room of 23 will share the same birthdate.

Let’s calculate the probability that 23 people in a room DO NOT share the same birthday. Starting with the first person, they have a unique birthday and it could be any birthday out of 365 days, so the probability of their birthday is 365/365 = 1. For the second person to have a birthday that IS NOT the same as the first person, their birthday could be the other 364 days of the year, or otherwise, 345/365. For the third person to not share the same birthday, there are 363 possible days for their birthday to land on.

To calculate the probability of all of the people not having the same birthday, we will have to multiply the probabilities together: 365/365 * 364/365 * 363/365 ….. = 0.4927 = 49.27%

Remember, this is the probability that the 23 people in a room do not share a birthday. To calculate the probability that they share a birthday, we need to take 100% – 49.27% = 50.73%! The birthday paradox is why you might notice that on Facebook, multiple of your friends likely share the same birthday!

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