Solution: 2016 Fall Midterm - 17

Author: Michiel Smid

Question

In the game of Hearthstone, you are given a deck of 18 distinct cards. One of the cards is the Raven Idol. You choose a uniformly random hand of 3 cards. Define the event

A = the hand of 3 cards contains the Raven Idol $

What is $\Pr(A)$?

(a)

3/18

(b)

3/19

(c)

3/17

(d)

4/19

Solution

Assume we obtain the Raven Idol first. Only once option.

For the other two cards, we choose 2 cards from the remaining 17 cards.

There are a total of $ \binom{17}{2} $ ways to do have an Idol Raven.

There are a total of $ \binom{18}{3} $ ways to choose 3 cards from 18.

$ Pr(A) = \frac{\binom{17}{2}}{\binom{18}{3}} $

$ Pr(A) = \frac{ \frac{17!}{15!2!}}{ \frac{18!}{3!15!}} $

$ Pr(A) = \frac{17!}{15!2!} \cdot \frac{3!15!}{18!} $

$ Pr(A) = \frac{17!}{2!} \cdot \frac{3!}{18!} $

$ Pr(A) = \frac{1}{2!} \cdot \frac{3!}{18} $

$ Pr(A) = \frac{1}{1} \cdot \frac{3}{18} $

$ Pr(A) = \frac{3}{18} $