Birthday problem

The birthday paradox reveals a surprising probability result in probability theory. It shows that in a group of 23 people, there's a 50% chance that at least two individuals share the same birthday. This counterintuitive fact challenges our initial assumptions about probability and randomness.

The paradox arises because we compare every possible pair of individuals in the group. With 23 people, there are 253 unique pairs to consider. As the number of pairs increases, the probability of at least one shared birthday also increases, eventually surpassing 50% at 23 people.

Real-world applications of the birthday paradox include cryptographic attacks like the birthday attack, which leverages this probabilistic model to find collisions in hash functions more efficiently. Understanding this paradox helps us grasp the underlying principles of probability and randomness in various fields.