The birthday problem can be formulated as follows:
In a class there are n students. What is the probability that at least two students were born on the same day of the year?
For simplify the math, without changing the result significantly, let's assume that year is always 365 days long (no February 29 birthdays) and let's assume that a person has an equal chance of being born on any day of the year, although some birthday may be slightly more likely than others.
If the students are 366 the probability would be equal to 1. If there is just 1 student the probability would be equal to 0.
Which is the n of student needed for the probability to be 0.5?
Most people would answer 183 and explain their answer saying that with this number of student the probability must be 0.5 because of this proportion:
366 : 1 = x : 0.5
This is the correct answer to a very different question: "How many student do you need in a class so that there is a 0.5 probability that one of them will share YOUR birthday?" If instead there are no restrictions on which two students will share a birthday, it makes a very big difference. If we take a class of 23 students there are 253 different ways of pairing two people together, and that gives a lot of possibilities of finding a pair with the same birthday.
To solve the birthday problem we need to use one of the basic rules of probability: the sum of the probability that an event will happen and the probability that the event won't happen is always 1. (In other words, the chance that anything might or might not happen is always 100%). If we can work out the probability that no two people will have the same birthday, we can use this rule to find the probability that two people will share a birthday:
P(event happens) + P(event doesn't happen) = 1
P(two people share birthday) + P(no two people share birthday) = 1 P(two people share birthday) = 1 - P(no two people share...