# Birthday Problem redux

A former student (Elizabeth) sent me a note.  She asked,

“I was looking at a limited lineage of my family tree of my ancestors…. Out of 18 individuals and 365 days out of the year, only one date was repeated as a day these individuals were born, and it was repeated 3 times. The day was February 13. Can you calculate the probability of this event?”

There are MANY good math problems here.  I’ll start with Elizabeth’s direct question and offer some alternatives.  All of my answers use 365 days/year (ignoring leap years) and assume all birthdays are equally likely.  Both of these have some problems, but it makes the solutions a bit easier.

1. What is the probability of exactly 3 people out of 18 being born on February 13 and no others sharing a birthday?
Of the 18 relatives, pick the 3 who share the February 13 birthday.  This can happen in $\displaystyle C\left(18,3\right)=\frac{18!}{(18-3)!\cdot 3!}=816$ ways. The other 15 relatives have 15 different birthdays in the remaining 364 days of the year.  This can happen in $\displaystyle P\left(364,15\right)\approx 1.95\cdot 10^{38}$ ways (This is a big number!).  So the probability of Elizabeth’s exact query is $\displaystyle \frac{C\left(18,3\right) \cdot P\left(364,15\right)}{365^{18}}\approx 1.20\cdot 10^{-5}$ , or a little more than 1 chance in 83,333.
2. What is the probability of exactly 3 people out of 18 being born on the same day and no others sharing a birthday?
The difference now is that the shared birthday could be anywhere in the calendar.  There are 365 ways that could happen with the rest of the problem being the same. This new probability is $\displaystyle \frac{365\cdot C\left(18,3\right) \cdot P\left(364,15\right)}{365^{18}}\approx 0.00439$ , roughly 1 chance in 278.
3. What is the probability that nobody in a group of 18 people shares a birthday?
That means choosing 18 of the 365 birthdays.  This probability is $\displaystyle \frac{P\left(365,18\right)}{365^{18}}\approx 0.653=65.3\%$.

Perhaps much more interesting than the last question (to me anyway) is that this means there is a $1-0.653=34.7\%$ chance that there are shared birthdays in any group of 18 people–a surprisingly high percentage to most people.  This is closely related to the “famous” birthday problem for which there are MANY online explanations (e.g., here and here).

QUICK MATH TRIVIA:  In any random group of 22 or more people, there is at least a 50% chance that two people will share the same birthday.  Thanks to knowledge of the birthday problem, this could win you some surprisingly easy bets!

If you want to read more about the probabilities of triplets of birthdays, I suggest this Math Forum post.  Make sure you read the last entry by Doctor Rick; the first answer is not quite correct.

Wolfram Alpha’s discussion of the Birthday Problem moves very quickly to a generalized solution if you can stomach lots of symbols. The complexity of the computations mentioned in the WA article are necessary if you consider the number of ways birthday pairings could occur.  In all cases, birthday problem solutions almost always boil down to computing the number of ways for an event to NOT occur and subtracting the result from 1.

Combinatorics problems like these are deceptively simple to state in words and are often notoriously complicated to state mathematically.  Once you have a good framing of the problem, though, they can be straightforward to compute, even if cumbersome.

BIAS NOTE:  All of these computations assume all birthdays are equally likely.  This simply isn’t true. Roy Murphy’s analysis (here) concludes the following.

The months July – October show higher than expected births and March – May show the most significant decline in births. Perhaps the most reasonable explanation is that conceptions are up in the months of October through January and down in June through August. You be the judge.

I thought this infographic was an especially cool demonstration: Great mathematics problems just keep on giving.