# The “Shared Birthday” Problem

“If a room has 23 people in it, what is the approximate chance that at least one pair of people in the room share the same birthday?”

This question is similar to GMAT math questions that test your knowledge of probability theory.  It requires too much calculation to actually be used as a GMAT question, but the technique for solving it is exactly the same.

Often, it’s easier to calculate the probability of a set of events NOT happening than it is to calculate the probability of that set of events actually occurring.

We can use the fact that, for any given set of events, the probability that it WILL occur plus the probability that it will NOT occur equals 1:

• P (birthday is shared) + P (nobody shares a birthday) = 1

So, since it’s easier to solve for the probability that nobody shares a birthday, we can solve the equation for the probability that there is at least one shared birthday:

• P (birthday is shared) = 1 – P (nobody shares a birthday)

So, what is the chance that nobody shares a birthday?

Start with the chances of two people not sharing a birthday.  For the first person chosen, we choose from 365 choices of birthdays available: $\frac{365}{365}$, which equals a chance of 1.  For the second person, we have 364 remaining birthdays to choose from out of 365:  $\frac{364}{365}$.  Multiply the two together and you get a chance of $\frac{364}{365}$, or about 99.7%, that those two people have different birthdays, and $\frac{1}{365}$, or 0.3%, as the chance that they share a birthday.

For each person added, we have one fewer birthday to choose from (363, 362, 361, etc.) out of the 365 days; multiply them all together to get the result:

• $\frac{365}{365}\frac{364}{365}\frac{363}{365}\frac{362}{365}...\frac{343}{365} =$

Do all the painful math (a spreadsheet helps a lot, unless you are a math savant), and you get about 49.3% chance that a birthday is NOT shared; subtract from 100% and you have a 50.7%, or about 1 in 2, chance that a birthday is shared between at least two people in the room.

So, next time you’re at a party and you count 23 people, it’s a fun bet to make that there will be a shared birthday.  32 people brings the odds up to 75%, and 40 people puts it right about at 90%.

(Yes, I know we’re ignoring the poor folks who were born on February 29 and only have a quarter as many birthdays to celebrate as the rest of us, but they change the odds only a tiny amount.)

If you enjoy math problems like this, you would probably enjoy, and do well on, the GMAT test.  Contact Bobby Hood Test Prep and I’d be happy to discuss the various class and tutoring options for the GMAT, both locally in Austin and online through The Princeton Review’s LiveOnline classrooms.