Mathematical Methods in the Physical Sciences · Problem 4.10

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · birthday-probability

Problem

What is the probability that you and a friend have different birthdays? (For simplicity, let a year have 365 days.) What is the probability that three people have three different birthdays? Show that the probability that

n

people have

n

different birthdays is

p = \left(1 - \frac{1}{365}\right)\left(1 - \frac{2}{365}\right)\left(1 - \frac{3}{365}\right) \cdots \left(1 - \frac{n-1}{365}\right).

Estimate this for

n \ll 365

by calculating

\ln p

[recall that

\ln(1 + x)

is approximately

x

for

x \ll 1

]. Find the smallest (integral)

n

for which

p < \frac{1}{2}

. Hence show that for a group of 23 people or more, the probability is greater than

\frac{1}{2}

that two of them have the same birthday. (Try it with a group of friends or a list of people such as the presidents of the United States.)

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