You say it’s your birthday…

Here’s a seemingly simple question: how many people do you have to have in a room for the chances of any two people having the same birthday (day and month only) to be greater than 50% (that is, 1/2)? We’re not talking about any specific birthday, just any birthday. Ignore leap years and assume that births are equally distributed throughout the year (both of these enormously complicate the math required, without significantly affecting the answer). I’ve asked this question at dinner parties before, and the guesses are generally somewhere around 183, which is just greater than half of 365, the total number of days in a year.

Most people’s immediate reaction is to think the likelihood of two random people sharing the same birthday is relatively low. However, this neglects one important considerations: multiple comparisons. Let’s answer the question:

Suppose Alice and Billy are in the room together. The chance of Alice and Billy having the same birthday is 1/365. Now suppose Carol walks in. What is the chance of a match now? Most people’s intuitive reaction is to say: the chance is now 2/365 (1/365 + 1/365). Naturally, that intuition is false (or else I wouldn’t be devoting web space to the problem). Let’s break down the odds:

Alice – Billy match = 1/365
Alice – Carol match = 1/365
Billy – Carol match = 1/365

In fact, the odds are now 3/365, not 2/365. “So what?”, you say; this is still very low. While that’s true, something strange has emerged. Let’s consider adding one more person, David.

Alice – Billy match = 1/365
Alice – Carol match = 1/365
Alice – David match = 1/365
Billy – Carol match = 1/365
Billy – David match = 1/365
Carol – David match = 1/365

Total = 6/365

In fact, the probability (p) of a match between “n” people (p(n), where n is some number of people) is given by the mathematical formula:

p(n) = 1 – 365!/365^n(365-n)!

The mathematical formula itself isn’t important here. What is critical to remember is that each new person who comes in the room (next comes Eleanor, then Fred, then Gregory) is a potential match with someone, and, in fact, the later one arrives to the room, the more “chances” of a match one brings. For instance, Carol’s arrival only brought two extra chances of a match, while David brought four.

In fact, after the 23rd person enters the room, the total number of possible matches will be greater than 183/365, making the probability of a match greater than 50%. Even more striking is that if we have 100 people in the room, the chances of a birthday match are far greater than 99% (we don’t reach 100% certainty until the 366th person enters). This is shown graphically below:

Birthdays

The point of all this is to emphasize that we every often misinterpret probabilities of things that we intuitively think are unlikely, without giving any thought to the real likelihood of the event. Once again, our intuition deceives. If we are fooled by our intuition at this point in our history, where science, logic, probability, and reason are so widely available, imagine how easy it must have been to fool our ancient brethren, who had no access to these faculties (or who were brutally oppressed lest they criticize accepted doctrine). Food for thought, I think…

I remain,

Michael

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