Question

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3},$ and each of the other five sides has probability $\frac{1}{15}.$ Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$

Answer

Answer: 167

Concepts:conditional probability Bayes’ Theorem dice (probability)

Difficulty rating: 2390

Solution:

The desired conditional probability is $\begin{aligned} &\frac{\Pr(\text{three sixes})}{\Pr(\text{first two are sixes})} \\ &= \frac{\frac{1}{2}\left(\frac{2}{3}\right)^3 + \frac{1}{2}\left(\frac{1}{6}\right)^3} {\frac{1}{2}\left(\frac{2}{3}\right)^2 + \frac{1}{2}\left(\frac{1}{6}\right)^2}, \end{aligned}$ since each die is chosen with probability $\frac{1}{2}$ and the fair die shows a six with probability $\frac{1}{6}.$

The numerator is $\frac{1}{2}\left(\frac{8}{27} + \frac{1}{216}\right) = \frac{65}{432}$ and the denominator is $\frac{1}{2}\left(\frac{4}{9} + \frac{1}{36}\right) = \frac{17}{72},$ so the probability is $\frac{65}{432} \cdot \frac{72}{17} = \frac{65}{102}.$

Since $65 = 5 \cdot 13$ and $102 = 2 \cdot 3 \cdot 17$ share no factor, $p + q = 65 + 102 = 167.$

2014 AIME II Problem 6

Answer: 167

Difficulty rating: 2390

Solution:

Problem 6 in Other Years