Have you ever shared a birthday with someone you know, either at school or work? The chance that this occurs is a lot more common than we realize. As a matter of fact, the probability of sharing a birthday in a group of only 30 people (which is around the size of an elementary school classroom) is over 70%!
A few years ago, I posted about a concept by Douglas W. Hubbard known as the "rule of five", which he discussed in his book, How to Measure Anything. At that time, I mentioned a famous problem in probability and statistics known simply as "the birthday problem", which seems counterintuitive until you actually take the time to work out the mathematics. Let's take a closer look.
Let's first review a couple of rules of simple probability. Remember that if you are rolling a standard die, the chance of coming up with the number "3" is 1 chance out of six, or 1/6. Similarly, the chance of rolling an even number (2, 4, or 6) is 3 chances out of six, or 3/6 (which of course simplifies into 1/2 or 50%). What happens if you roll two dice? What is the chance that you will roll "3" with the first die and "4" with the second? The simple rules of probability state that the chance of two independent events occurring is a product of their individual probabilities. In other words, we multiply 1/6 (the chance of rolling a "3") by 1/6 (the chance of rolling a "4"): 1/6 x 1/6 = 0.03, or 3%. What's the probability of not rolling a "3" on one die? It's just 1 minus the probability of rolling a "3", or 1 - 1/6 = 5/6. You now have all the tools to answer the birthday problem!
Suppose you are a student in a classroom with 29 other students. What is the probability that you share a birthday with at least one other student in the class? The best way to tackle this problem is to first calculate the probability (P) that these 30 individuals do NOT share a same birthday, P(A). Therefore, according to the simple rules of probability discussed above, the probability that at least one student shares his or her birthday with someone else in the class is P(B) = 1 - P(A).
Let's also make things a little more straightforward and assume that there aren't any twins in the class (twins would share the same birthday) and that there are no leap years involved, which means that there are 365 possible birthdays in a year. Imagine that we ask all 30 students to form a line, and each student will go and mark his or her birthday on a calendar, one right after the other. The probability that the first student will have a birthday that is not shared with anyone else is 365/365. The probability for the second student not sharing a birthday with the first student is 364/365. The probability for the third student not sharing a birthday with the first or second students is 363/365. These are all independent events, so in order to calculate the probability of all 30 students not sharing a birthday, we have to multiply the probability of the individual students:
P(A) = 365/365 x 364/365 x 363/365 x ... 337/365 x 336/365 = 0.293684
The probability that at least two individuals share the same birthday, P(B), is therefore:
P(B) = 1 - P(A) = 1- 0.293684 = 0.706 or 70.6%!
If you look the "birthday problem" up online, you can find a graph of P(B) as a function of the number of people in a group that looks something like this:
The "birthday problem" is a great example of what is called a veridical paradox, which is something that appears to be false but is nonetheless true. Next time, we will discuss another famous veridical paradox (and I promise I will explain what all of this has to do with leadership next time too).
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