In a group of 23 people, the probability that 2 of them share a birthday is 50%. This fact may seem strange at first, but can be proven by fairly simple mathematics.
In a group of 23 people, there are 253 unique pairings which can be made. This can be calculated by finding 23C2.
Assuming that each day in the year has an equal probability of being somebody’s birthday, the probability two people share a birthday will be 1/365.
This means the probability that they do not share a birthday will be 364/365. For nobody in the group of 23 to have a shared birthday, all 253 pairings will have to not share a birthday. As this is an AND probability, all these probabilities must be multiplied together.
Therefore, the probability that nobody in the group shares a birthday is:
P = (364/365)253 = 0.4995
Overall, the probability that two or more people have a shared birthday will be just over 50% (0.5005)
What if there are 75 people?
Applying the same method, it can be shown that there is a 99.95% chance that 2 people share a birthday out of a group of 75.
75C2 is equal to 2775.
1-(364/365)2775 = 0.9995