When I decided to become a mathematics major in college, I knew that in order to complete this degree, two of the required courses–besides advanced calculus–were Probability Theory and Math 52, which was statistics. Although probability was a course I was looking forward to, given my penchant for numbers and games of chance, I quickly learned that this theoretical math course was no walk in the park. This notwithstanding, it was in this course that I learned about the birthday paradox and the mathematics behind it. Yes, in a room of about twenty-five people the odds that at least two share a common birthday are better than 50-50. Read on and see why cours particuliers maths.
The birthday paradox has to be one of the most famous and well known problems in probability. In a nutshell, this problem asks the question, “In a room of about twenty-five people, what is the probability that at least a pair will have a common birthday?” Some of you may have intuitively experienced the birthday paradox in your everyday lives when talking and associating with people. For example, do you ever remember talking casually with someone you just met at a party and finding out that their brother had the same birthday as your sister? In fact, after reading this article, if you form a mind-set for this phenomenon, you will start noticing that the birthday paradox is more common than you think.
Because there are 365 possible days on which birthdays can fall, it seems improbable that in a room of twenty-five people the odds of two people having a common birthday should be better than even. And yet this is entirely the case. Remember. The key is that we are not saying which two people will have a common birthday, just that some two will have a common date in hand. The way I will show this to be true is by examining the mathematics behind the scenes. The beauty of this explanation will be that you will not require more than a basic understanding of arithmetic to grasp the import of this paradox. That’s right. You will not have to be versed in combinatorial analysis, permutation theory, complementary probability spaces–no not any of these! All you will need to do is put your thinking cap on and come take this quick ride with me. Let’s go.
To understand the birthday paradox, we will first look at a simplified version of the problem. Let’s look at the example with three different people and ask what the probability is that they will have a common birthday. Many times a problem in probability is solved by looking at the complementary problem. What we mean by this is quite simple. In this example, the given problem is the probability that two of them have a common birthday. The complementary problem is the probability that none have a common birthday. Either they have a common birthday or not; these are the only two possibilities and thus this is the approach we will take to solve our problem. This is completely analogous to having the situation in which a person has two choices A or B. If they choose A then they did not choose B and vice versa.
In the birthday problem with the three people, let A be the choice or probability that two have a common birthday. Then B is the choice or probability that no two have a common birthday. In probability problems, the outcomes which make up an experiment are called the probability sample space. To make this crystal clear, take a bag with 10 balls numbered 1-10. The probability space consists of the 10 numbered balls. The probability of the entire space is always equal to one, and the probability of any event that forms part of the space will always be some fraction less than or equal to one. For example, in the numbered ball scenario, the probability of choosing any ball if you reach in the bag and pull one out is 10/10 or 1; however, the probability of choosing a specific numbered ball is 1/10. Notice the distinction carefully.