The Birthday Paradox : Probability and Statistics

Before we start this video, let's ask you a question. If there are 23 randomly chosen people in a room, what is the probability that any two of them have their birthday on the same day? You may think that number should be very low because each person's birthday might be randomly falling into any day within 365 days in a year. However, if you do a serious math calculation, you may find that the actual result is 50.7%, which might be much higher than you originally expected, isn't it? This is what the birthday paradox is talking about, counterintuitively, the probability of a shared birthday exceeds 50% in a group of only 23 people. A more surprising result is that if the number of people is increased to 50 in the room, the probability that two people share the same birthday will be increased to 97%.

This birthday paradox is a famous veridical paradox, which means that it appears absurd, but is actually to be true. Let's illustrate the mathematical calculation of the probability. Let's number the 23 randomly chosen people from number 1 to 23. We will ask them to enter the room one by one, and we will calculate the probability that the newly entered person doesn't share the birthday with all people already in the room. When the first person enters, since there are no people in the room, the probability that he didn't share the birthday with people in the room is 100%. When person 2 enters the room, since person 1 is already in the room, the probability that he didn't share the birthday with people in the room is 364/365. Similarly, when person 3 enters the room, his probability of not sharing a birthday with other people is 363/365, and so on. If we multiply those probability together, we can calculate the final probability that no people in the room share the birthday with another in this room, and the following table shows the exact probability based on the number of people.

Probability = 1 x 364/365 x 363/365 ......

number of people in the room probability of two people having the same birthday
10 12%
20 41%
30 70%
50 97%
100 99.99996%
200 99.9999999999999999999999999998%
300 1 −（7×10−73）
350 1 −（3×10−131）
≥366 100%

From the data in the table, we can see if there are only 10 people in the room, the chance to have two people sharing the birthday is low, which is just about 12%. However, when there are more than 50 people in the room, the final probability can reach more than 97%. The reason that most people intuitively think that the probability that 2 out of 23 people have the same birthday should be much less than 50% is because they may mis-understand the problem as "the probability that the other 22 people have the same birthday as you". If you notice that the question is asking "the probability that any two people share the same birthday", you may realize that the result will be much more than you previously thought.