Solution: 2018 Fall Final - 13

Author: Michiel Smid

Question

You are given a fair die that has six faces. One face has the letter $a$ on it, two faces have the letter $b$ on them, and three faces have the letter $c$ on them. Assume you roll this die twice, independently of each other. Define the events

A = "both rolls result in the same letter",
B = "at least one of the rolls results in the letter $a$".

What is $\Pr(A|B)$?

(a)

2/7

(b)

2/11

(c)

1/11

(d)

1/7

Solution

Let's determine $ B $
What we care about is when we get an $a$ so let's map that out with a table

$ |B| = 11 $
$ Pr(B) = \frac{11}{36} $
Let's determine $ A \cap B $
We only care about the cases when both rolls result in the same letter AND at least one of the rolls result in the letter $a$
![alt text](image-1.png)
$ |A \cap B| = 1 $
$ Pr(A \cap B) = \frac{1}{36} $

$ Pr(A|B) = \frac{ Pr(A \cap B) }{ Pr(B) } $

$ Pr(A|B) = \frac{ \frac{1}{36} }{ \frac{11}{36} } $

$ Pr(A|B) = \frac{1}{11} $