How many people do you need in a room until there are two people with the same birthdays? The pigeonhole principle dictates that (excluding February 29) since there are 365 birthdays, 366 people in a room would guarantee two people sharing birthdays. However, this is only the number needed to absolutely guarantee a pairing. Using a neat statistical trick known as the birthday problem (or birthday paradox), we can find that a much smaller number is needed to solve the problem.
Let us assume that every birthday is equally possible (in real life, some birthdays are more common than others). If there are 30 people in the room, Person 1 has a chance of sharing a birthday with each of the other 29 people (possible pairs). Person 2 can be paired with 28 people (since they have already been “paired” to 1), Person 3 with 27 people and so forth. Therefore, the number of chances are: 29 + 28 + 27+ … + 1. Using Gauss’s handy addition trick, the total number is (29 + 1) x 29/2 = 435. We can see already that although the total number of individuals is only 30, the total number of pairs already exceeds 365. Since the probability of having a certain birthday is 1/365, it is likely that it would occur when you have so many possible chances.
Using statistical analysis, it can be found that when there are 23 people, the odds of there being a match surpasses 50%, making it more likely that two people share a birthday than not. By 70 people, the probability of a match grows to 99.9%. Therefore, with only 19% of the number required by the pigeonhole principle, the birthday problem can say with 99.9% certainty that there will be two people sharing a birthday.
If I have three gloves, there must be at least either two left gloves or two right gloves. It is impossible to have one left glove, one right glove and a third glove that is neither left nor right (usually). This logic is called the pigeonhole principle. It is named because of the logic that if you have n pigeons and m pigeonholes where n > m (e.g. 10 pigeons in 9 holes), then at least one pigeonhole must contain more than one pigeon. This is because the biggest spread of the pigeons is putting at least one in each box, but as n > m, there is a pigeon left over and it must go in a box with another pigeon. The pigeonhole principle seems like a basic counting principle, but its implications are quite interesting.
For example, let’s say that your sock drawer is very unorganised and has a mix of black and white socks. What is the minimum number of socks you need to pick out before you get two of the same colour? The pigeonhole principle dictates that when n > m, each “slot” must be filled with more than one item. Here, the slot is colour. As there are two colours (m = 2), you only have to pick three socks out to have a matching pair (n = 3, 3 > 2).
The pigeonhole principle allows us to make seemingly impossible conjectures, such as the fact that a person living in London will have the exact number of hairs on their head as at least one other person living in London. An average human head has about 150,000 hairs and it would be a safe assumption to say that no one would have more than a million hairs on their head (m = 1,000,000). The population of London far exceeds a million (n > 1,000,000), therefore, there must at least two people living in London with the exact same amount of hair on their head. Similarly, if you are in a room with 366 other people, you are guaranteed to share a birthday with at least one person.
The ritual of blowing out candles on one’s birthday is interesting as it shows the characteristics of human beings very well. This ritual shows that the person can make fire while reminding themselves they can extinguish it with one breath. It is a ritual that helps a baby develop into a responsible, social being that is capable of controlling fire. On the other hand, an old person being so breathless that they cannot even blow out a candle signifies that it is time for them to be socially excluded by the active population.