The Birthday Paradox and the Pigeonhole Principle
The Birthday Paradox and the Pigeonhole Principle are two mathematical principles that are closely related. Both are related to statistics, probability, and combinatorics. The Birthday Paradox states that in a group of at least 23 randomly chosen people, there is at least a 50% chance that two people will have the same birthday either in the same calendar year or a different one. The Pigeonhole Principle states that if n items are to be put into m containers, and n is greater than m, then at least one container will contain two or more items.
The Birthday Paradox is an example of a Pigeonhole Principle. Lets use the example of a group of 30 randomly chosen people. Here, there are 365 days in the year for each person in the group to have a unique birthday, and therefore we have 30 different pigeonholes and three hundred and sixty five different items (the days of the year). Since 30 is less than 365, this is a classic example of the Pigeonhole Principle. As such, at least two people in the group must have the same birthday, and the probability of this happening is exactly 50%.
This concept can be extended to include people in more than one calendar year. In this case, there can be 730 different items (the days of two calendar years combined) and if n (the number of people in the group) is greater than 730 then the Pigeonhole Principle will apply, and at least two people in the group may share the same birthday.
The Birthday Paradox is a counter-intuitive result with many applications in computer science. It can be used to measure the security of some cryptographic systems and can be used to demonstrate the unlikely nature of mathematical coincidences. It is also used to explain why lottery numbers are picked so as to avoid a combination of numbers that represent a persons birthday.
The Birthday Paradox also demonstrates how a small change can have a large effect. For example, if the group of 30 people is expanded to 43, the chance of two people having the same birthday increases to over 97%.
The Pigeonhole Principle is a more general concept. It is often used in mathematics, computer science, and many other fields. In the case of the Birthday Paradox, it is used to explain why two people might have the same birthday. This principle can also be used to explain why people must have enough different passwords for all their online accounts, or why it is unlikely to have a successful lottery ticket with six numbers that are all the same digits of a persons date of birth.
In conclusion, the Birthday Paradox and the Pigeonhole Principle are closely related mathematical concepts with many applications in mathematics, computer science, and other fields. The Birthday Paradox explains why two people in a group are likely to have the same birthday, while the Pigeonhole Principle is a more general concept about the limitations of putting items into containers. Together, these two principles can help us better understand the nature of probability and coincidence.
