11. Billie has a die with the numbers $1,2,3,4,5$ and 6 on its six faces.
Niles has a die which has the numbers $4,4,4,5,5$ and 5 on its six faces.
When Billie and Niles roll their dice the one with the larger number wins. If the two numbers are equal it is a draw.
The probability that Niles wins, when written as a fraction in its lowest terms, is $\frac{p}{q}$. What is the value of $7 p+11 q$ ?
SolUTION
181

Niles rolls a 4 with probability $\frac{1}{2}$, and then wins if Billie rolls a 1,2 or 3 , with probability $\frac{1}{2}$. Niles rolls a 5 with probability $\frac{1}{2}$, and then wins if Billie rolls a $1,2,3$ or 4 , with probability $\frac{2}{3}$. Therefore, the probability of Niles winning is $\frac{1}{2} \times \frac{1}{2}+\frac{1}{2} \times \frac{2}{3}=\frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}$. Therefore $p=7, q=12$ and $7 p+11 q=7 \times 7+11 \times 12=181$.