The Birthday Paradox by James Storey (8R)
At a party what are the chances that a pair of people share the same birthday? This is known as the Birthday Problem. It is a probability problem with a surprising solution!
We need to define the problem more clearly in order to get an answer. Letβs say that there was a party and you were the first person to arrive. Clearly, there would be a 100% chance that your birthday matches your birthday! So far, easy!
Now another person walks into the room, what is the probability of them sharing your birthday?
Well, letβs consider the chances of them not sharing your birthday. There are 365 days in a year (not including leap years) and so the other person therefore has a 364 365 chance of not sharing your birthday. This is a 99.7% chance of them not sharing your Birthday which leaves a 0.3% chance that they will share your birthday.
Now assuming that another person walks in there is a 363 365
of them not sharing a birthday with the two
of you in the room providing that the two people in the room do not share the same birthday. So we now have the probability of not sharing a birthday as
365 365
π 364 365
π 363 365
π 100 = 99.179%
So the probability that they share a birthday with at least one person in the room is 100 β 99.179 = 0.821%.
Applying this as more people enter we get an interesting effect ... it only requires 23 people to enter the room before there is a 50% chance of two people sharing the same birthday.
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