FILE No.80Unsolved
At a puzzle club meeting, a detective says: "I bet there is a better than even chance that two of us share a birthday." Birthdays are spread evenly over 365 days. Ignore leap years.
Someone replies, "A year has 365 days, so you would need about half of that, 183 people."
What is the smallest number of people for which the chance of a shared birthday exceeds 50%?
If you get stuck, open the hints one at a time, in order.
Work out the chance that everyone differs as you add people.
How many pairs of people are there among 23?
At 23 people, is "everyone differs" above or below half?
What is the smallest number of people for which the chance of a shared birthday exceeds 50%?
Choose the answer closest to the truth.
23
The answer is: 23.
Multiply the chances that everyone has a different birthday as you add people: (365/365) x (364/365) x (363/365) x ... With 23 people, the chance everyone is different is about 49.3%, so the chance someone matches is about 50.7%. With 22 people it is about 47.6%, which is still under half.
The number of pairs grows fast. 23 people form 253 different pairs.
People compare 365 days with the number of people and expect about 183.
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