The Birthday Problem

Due to some scheduling conflicts, I am delaying the publication of part II of my series on 3D printing until April the 7th. In the meantime, please enjoy this short column on a common math puzzle.

You may have heard that if you are in a room with 23 people, there is a greater than 50% chance that two people in the room have the same birthday. This is commonly known as the “Birthday Problem.” Most people presented with this information are, at least initially, quite skeptical. Typically what throws people off is that they approach the problem with an incorrect perspective.

The problem stems from considering the Birthday Problem from the stand point of “how likely is it for someone in the room to have the same birthday as me?” Since there are 365 days in the year, one more than half is 183. Therefore, the number of people in the room required such that chances of someone having the same birthday as you are greater than 50% is 183. As long as you are thinking about the problem that way, the number of 23 sounds preposterously low. The key to understanding the math behind the Birthday Problem is that it is not addressing whether someone in the room has the same birthday as you, but rather whether any two people in the room share the same birthday.

The mathematics for the Birthday Problem are sort of fun, at least if you are a nerd like me. It’s easier to work through the problem from the reverse angle of asking, “What are the chances that no two people it the room have the same birthday?” Let’s start working through the problem one person at a time.

• If there is only one person in the room, then the probability of there not being two people with the same birthday is 100%.
• Now a second person enters the room. Ignoring leap years, there is a 364/365 chance that the second person does not have the same birthday as the first. So the probability that they do have the same birthday is 1-(364/365) = 0.003 or 0.3%.
• When the third person enters the room, there is a 363/365 that she does not have the same birthday as either of the first two people. Therefore, the overall probability that any two of the three people in the room do share a common birthday is 1 – (364/365)*(363/365) = 0.008 or 0.8%.
• The pattern continues when the fourth person enters the room,such that the probability that any two of the four do share a common birthday is now 1-(364/365)*(363/365)*(362/365) = 1.6%.

As you continue this mathematic series, the chance that any two people in the room share the same birthday exceeds 50% when the 23rd person enters. When the 70th person enters the room, the chance that two people share the same birthday reaches 99.9%, a near guarantee.

While this example is just sort of a fun number puzzle, the difficulties we all share when confronted with numbers and probabilities affect our personal lives and governmental policies in profound ways, in particular in assessing risks. I’m tempted to expound on that further, but I promised a short column, so for now just enjoy the numbers.

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