Question

Assuming that all years have 365 days and all birthdays occur with equal probability, how large μst n be so that in any randomly chosen group of n people, the probability that two or more have the same birthday is at least 1/2? (This is called the birthday problem. Many people find the answer surprising.)

a) 15
b) 20
c) 23
d) 30

Answer 1

In the birthday problem, at least 23 people are needed in a group to have a 50% or greater chance that two people share the same birthday, considering all days are equally likely for a birthday and a year has 365 days.

Step-by-step explanation:

The birthday problem asks for the minimum number of people, n, required in a group so that there is at least a 50% chance that two people share the same birthday. To find this number, one way is to calculate the probability that all n people have different birthdays and subtract it from 1. The problem states that we assume each year has 365 days and that all birthdays are equally likely.

Starting with the first person, they can have any birthday, so the probability of a unique birthday is 365/365. The second person must have a different birthday, so their odds are 364/365. This continues diminishing with each additional person added. The probability that n people all have different birthdays is:

P(different) = (365/365) * (364/365) * ... * (365-n+1)/365

We want the probability of at least one shared birthday to be at least 1/2, so:

1 - P(different) ≥ 1/2

It is known from mathematical calculations and simulations that the smallest n that satisfies this is 23. Therefore, in a group of 23 people, there is at least a 50% chance that two people share the same birthday.