Written by Alan Bellows on 09 February 2006

Birthday Cake

I have never had a very good relationship with Mathematics. I used to think it was me… I thought that perhaps I was just a bit put off by Math’s confident demeanor and superior attitude, and by its tendency to micromanage every tiny detail of my universe. But over time I have come to the realization that I’m not the source of the problem. Math, as it turns out, is out of its bloody mind.

Consider the following example: Assuming for a moment that birthdays are evenly distributed throughout the year, if you’re sitting in a room with forty people in it, what are the chances that two of those people have the same birthday? For simplicity’s sake, we’ll ignore leap years. A reasonable, intelligent person might point out that the odds don’t reach 100% until there are 366 people in the room (the number of days in a year + 1)… and forty is about 11% of 366… so such a person might conclude that the odds of two people in forty sharing a birthday are about 11%. In reality, due to Math’s convoluted reasoning, the odds are about 90%. This phenomenon is known as the Birthday Paradox.

If the set of people is increased to sixty, the odds climb to above 99%. This means that with only sixty people in a room, even though there are 365 possible birthdays, it is almost certain that two people have a birthday on the same day. After making these preposterous assertions, Math then goes on to rationalize its claims by recruiting its bastard offspring: numbers and formulas.

It’s tricky to explain the phenomenon in a way that feels intuitive. You can consider the fact that forty people can be paired up in 780 unique ways, and it follows that there would be a good chance that at least one of those pairs would share a birthday. But that doesn’t really satisfy the question for me, it just feels marginally less screwy. So I did something quite out of character: I crunched the numbers. The values rapidly become unmanageable, but the trend is clear:

# of people Possible combinations of birthdays # of those combinations where at least two birthdays fall on the same day % of combinations where two people have same birthday
1 365 0 0.0%
2 133,225 365 0.2%
3 48,627,125 398,945 0.8%
4 17,748,900,625 290,299,465 1.6%
5 6,478,348,728,125 175,793,709,365 2.7%
6 2,364,597,285,765,625 95,677,479,012,025 4.0%
7 863,078,009,304,453,125 48,535,798,679,910,725 5.6%
8 315,023,473,396,125,390,625 23,417,361,992,539,211,425 7.4%

Only calculating up to eight people, we see that of the three hundred fifteen quintillion possible combinations of birthdays the group has, 7.4% of cases– or about one in thirteen– result in two of them having the same birthday. As each person is added, the odds do not increase linearly, but rather they curve upwards rapidly. This trend continues up to around twenty-three people, where the curve hits 50% odds, and the rate of increase starts going down. It practically flattens out when fifty-seven people are considered, and the odds rest at about 99%. Though it may not be intuitive, the numbers follow the pattern quite faithfully.

So does this mean that you can walk into a math class of forty students, bet them that at least two people in the room share a birthday, and win 90% of the time? Not exactly. In real life, where Math is not particularly welcome, birthdays are not distributed perfectly throughout the year. More people are born in the springtime, which throws the numbers off. Also, as a result of the way that hospitals operate, more babies are born on Mondays and Tuesdays than on weekends, which further complicates the problem. Depending on the group of people and how evenly distributed their birthdays are, the results can vary widely. But most of the time, you’ll still have some very good odds.

Birthday curveBut there is at least one highly practical application for this numerical phenomenon: computer hacking. There is a classic cryptographic computer attack known as the “birthday attack” which exploits the math of the birthday paradox. Using this method, a programmer can store the results of the birthday math in memory to decrease overall processing time when doing certain computationally useful things, such as attempting to crack a digital signature.

Another thing that I discovered in my research is that a one followed by fifty-one zeros is called one sexdecillion. I knew those mathematician guys were hiding something in those big numbers.

As much as Math would like us to think that it is an advocate for structure and intuition, every once in a while it churns up something dastardly and unintuitive like the Birthday Paradox, the Monty Hall problem or Benford’s Law. And we have no choice but to obey these fickle whims of the great control freak. But every once in a while, I like to divide by zero, just to show Math that I’m not powerless to retaliate.

To those who would claim that only a fool would fall prey to the Birthday Paradox, and that the true nature of the odds is perfectly intuitive, I ask this of your Rainman-like grasp of numbers… why is it that all of the totals in this article’s first chart (aside from zero) end in the digit five? That outcome surprised me, but I currently lack the conviction to pursue the matter. I now see that numbers represent all that is soulless and wrong.

Further reading:
Wikipedia article on the Birthday Attack
MathWorld description of the Birthday Problem