Solution: 2018 Fall Midterm - 16
Author: Michiel SmidQuestion
Alexa, Tri, and Zoltan each have a uniformly random birthday. (We ignore leap years, so that one
year has 365 days.)
Define the event
Define the event
- A = "Alexa, Tri, and Zoltan have different birthdays".
(a)
$\frac{362 \cdot 363 \cdot 364}{365^3}$
(b)
$\frac{363 \cdot 364}{365^2}$
(c)
$\frac{363^3}{362 \cdot 363 \cdot 364}$
(d)
$\frac{365^2}{363 \cdot 364}$
Solution
Alexa has 365 possible birthdays.
Tri has 364 possible birthdays since Tri can’t have the same birthday as Alexa.
Zoltan has 363 possible birthdays since Zoltan can’t have the same birthday as Alexa or Tri.
Thus, there are $ 365 \cdot 364 \cdot 363 $ ways for Alexa, Tri, and Zoltan to have different birthdays.
There are $ 365^3 $ ways for Alexa, Tri, and Zoltan to have birthdays.
Thus, Pr$ (A) = \frac{365 \cdot 364 \cdot 363}{365^3} = \frac{364 \cdot 363}{365^2} $