Birthday Sharing Odds

Your Birthday | Other People's Birthdays

Your Birthday

Let's say today's your birthday. I know it probably isn't. There's only about a 1 in 365 chance that today's your birthday. But, let's say it's your birthday today anyway, and you're having a party. While at the party, one of your guests tells you "Hey, guess what? Today's my birthday too."

You'd think that was pretty surprising -- coincidental even. "What are the chances!?" you might exclaim.

In general, the chances are slim. But, the chances also depend on how many people are at your party. You might liken the situation to rolling a die that has 365 sides (ignoring leap years). For each of your guests, you roll the die. Rolling a particular number (i.e., your birthday) is fairly dificult, but you probably appreciate that, as you role more and more, you're more likely to hit your number. The more guests you have at your party, the more likely it is that one of them shares your birthday.

Now, because you've bothered to read this far, I'll assume you're interested in answering the following question(s):

How many people would have to be at your party before you had a 50% chance that someone there shared your birthday?

or put another way:

How big of a party do you need to be in to take an even-money bet that someone at the party shares your birthday (and stand a more than fair chance of winning)?

To answer this (these) questions, let's first make some assumptions:

All years have 365 days (we'll ignore leap years)
All birthdays are equally likely (this is not acually true. Birthrates vary slightly by season).
You have no prior knowledge of your guests' birthdays (if you invite someone who you already know shares the same birthday as you... well, there's not much point in doing this exercise, now is there?)

If only one person shows up to your party, there's a 1 in 365 chance (~0.3%) that they share your birthday. Or rather, there's a 99.7% (364 in 365) chance that they do not share your birthday.

If two people show up to your party, the odds are ~0.5% that at least one of them shares your birthday. This is obtained by multiplying the chances that each person doesn't share your birthday and then subtracting that product from 100%.

In words:
Probability at least one of your two guests shares your birthday =
(100% of the time) - [ probability (that guest #1 doesn't share your birthday) and (that guest #2 doesn't share your burthday either) ]

In numbers:

For three guests:

For four guests:

By now, the pattern should be pretty clear. For any number of guests (n), the odds of at least one of them sharing your birthday is

Note that this is not the same as simply adding up the 1 in 365 chances for all your guests. It's clear that this latter technique is flawed in the case of 365 guests, which yields a 100% chance that someone shares your birthday. That's wrong. At a party with 365 guests, someone shares a birthday with someone else, but that someone else is not neccessarily you.

In fact, you would need an infinite number of guests to be 100% confident that at least one of them shared your birthday. As you can see from graphing the probability (x) versus the number of party guests (n), you can never quite reach 100%; even at large numbers of guests. This makes sense because even in very large groups, there’s still some small chance that no one will share your birthday.

Getting back to the original problem, we can rearrange the equation for figuring out the probability that someone shares your birthday (x):

Solving for the number of guests (n) yields:

or by approximation:

To garner at least a 50% chance (x = 0.5) that at least one one guest has the same birthday as you? You'd need 253 guests.

Don't hold your breath. You're not that popular.

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