Today’s Fall Through the Cracks Friday feature comes to us from the world of math via the New York Times Opinion page and one of my Facebook friends. It deals with the mathematical question called the “birthday problem” which asks how many people would be enough to make the odds of any two of them having the same birthday at least 50-50?

This question appealed to me because my son and I have the same birthday (I’m older) and because most people think the answer is much higher than it actually is (see answer at bottom of this post). Take your best guess at the answer and then read this analysis of the problem from the NYT:

Suppose you have 3 pairs of pants and 5 shirts. (I realize you probably have more than this, but pretend you’re a math professor.) How many different outfits can you create? (And don’t worry if some of the shirts and pants don’t go too well together — remember, you’re a math professor!) Say you decide to wear your ratty blue jeans. Then with five shirts to choose from, that gives you 5 outfits right there. Or you could wear those nice polyester khakis you still have from your high school graduation. Combine them with any of the five shirts and that’s another 5 outfits. Finally, you could go casual and wear your Star Trek sweat pants along with any of the five shirts, creating 5 more outfits and bringing the total to 3 times 5, or 15, outfits in all.

That’s the combination principle in action: If you can make M choices of one thing (like 3 pairs of pants) and N choices of another (like 5 shirts), you can make M x N combinations of them both (15 outfits). The principle also extends to more than two things. If you want to top off your outfit with a stylish hat and you have 6 to choose from, you can create 3 x 5 x 6 = 90 ensembles of pants, shirts and hats.

Next, let’s apply this principle to a warm-up birthday problem featuring the first three United States presidents. Relative to the New Style (Gregorian) calendar, George Washington was born on Feb. 22, John Adams on Oct. 30, and Thomas Jefferson on April 13. Unsurprisingly, no matches. To figure out the odds of this happening by chance, we imagine alternate realities — all the possible combinations of birthdays that could have occurred — and then calculate what fraction of those combinations involve three distinct birthdays.

According to the combination principle, there are 365 x 365 x 365 combinations of three birthdays, since any day of the year is possible for each of the three presidents. To count how many of these combinations contain no matches, let Washington go first. He has all 365 days at his disposal. But once his birthday is fixed, he leaves Adams with only 364 choices to avoid a match, which in turn leaves Jefferson only 363. So, by the combination principle, there are 365 x 364 x 363 non-matching combinations of three birthdays, out of a total of 365 x 365 x 365 combinations altogether.

Hence the probability that all three birthdays differ is the ratio of these huge numbers:

 

 

which is about 0.9918 or 99.18 percent. In other words, it was almost a sure thing that none of the birthdays would match, as we’d expect by common sense.

To extend this result to four or more people, look again at the fraction above and savor the patterns in it. For three people the fraction has three descending numbers — 365, 364, 363 — in the numerator, and three copies of 365 in the denominator.  So for four people, the natural and correct guess is that the answer becomes

 

 

This expression is merely the fraction we found earlier for three people, multiplied by 362/365. Doing the arithmetic then gives 0.9836, or a 98.36 percent chance that four random people have four different birthdays. That means the probability that two or more of them share a birthday is about 1 – 0.9836 = 0.0164, or 1.64 percent.

Continuing in this way, ideally with the help of a spreadsheet, computer or online birthday problem calculator, we can crank out the corresponding probabilities for any number of people. The calculations show that the odds of a match rise sharply as the group gets larger. With 10 people, the odds are almost 12 percent; with 20 people, 41 percent. When we reach the magic number of 23 people, the odds climb above 50 percent for the first time, which is what we were trying to prove.

How did your guess compare to the actual answer of just 23 people?

To read more on this problem, including a birthday question-related clip from the Tonight Show with Johnny Carson, check out the full post.

Have a great weekend!