Solution: 2015 Fall Final - 15

Author: Michiel Smid

Question

You are given that:

The course COMP 9999 runs over a period of one year, starting on January 1 and ending on December 31. There is one lecture every day; thus, the total number of lectures is 365.
Dania and Nick take this course. Dania's birthday is on November 19. Nick's birthday is on December 3.
Professor G. Ruesome teaches the course. Professor Ruesome decides to have 20 quizzes during the year. For this, he chooses a uniformly random subset of 20 days; the quizzes will be on the 20 chosen days. (It is possible that there is a quiz on January 1.)

Determine $\Pr(A)$, where $A$ is the event

A = "There is a quiz on Dania's birthday and there is a quiz on Nick's birthday".

(a)

$1 - \left. {363 \choose 20} \middle/ {365 \choose 20} \right.$

(b)

None of the above.

(c)

$\left. {363 \choose 18} \middle/ {365 \choose 20} \right.$

(d)

$\left. {365 \choose 18} \middle/ {365 \choose 20} \right.$

Let’s assume that the birthdays of Diana and Nick are chosen

That means we have 363 days left to choose 18 days for the quizzes: $ \binom{363}{18} $

The total number of ways to choose 20 days for the quizzes is $ \binom{365}{20} $

$ Pr(A) = \frac{\binom{363}{18}}{\binom{365}{20}} $