The Birthday Problem:
To solve the birthday problem, you need to find the probability that two people don't have the same birthday. Then you use the compliment (1-P(two people don't have the same birthday)). Then because of exponential growth, the chances of getting a match increase. Exponential growth rapidly decreases the chance of two people not having the same birthday, and therefore the chances of having the same birthday increases. The reason that when you have 23 people, the chances of having the same birthday are about 50% because the square root of 365 is about 20, meaning that about 50% of the time when there are about 20 people, two people will have the same birthday.
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