Solution: 2014 Fall Final - 13

Author: Michiel Smid

Question

Annie, Boris, and Charlie have random and independent birthdays. (We ignore leap years, so that a year has 365 days.) What is the probability that Annie, Boris, and Charlie have the same birthday?

(a)

$\frac{1}{364 \cdot 365}$

(b)

$\frac{365}{364^{2}}$

(c)

$\frac{1}{365^2}$

(d)

$\frac{1}{365^{3}}$

Solution

Let A = Event that Annie, Boris, and Charlie have the same birthday

Annie has a birthday on any of the 365 days: 365

Boris would only then have a birthday on the same day as Annie: 1

Charlie would only then have a birthday on the same day as Annie and Boris: 1

$ |A| = 365 \cdot 1 \cdot 1 $

Let S = All possible outcomes

$ |S| = 365^3 $

$ Pr(A) = \frac{365}{365^3} $

$ Pr(A) = \frac{1}{365^2} $