MathDB

Blue rolls a fair

n

-sided die that has sides its numbered with the integers from

1

n

, and then he flips a coin. Blue knows that the coin is weighted to land heads either

\dfrac{1}{3}

\dfrac{2}{3}

of the time. Given that the probability of both rolling a

7

and flipping heads is

\dfrac{1}{15}

, find

n

.
Proposed by Jacob Xu
Solution.

\boxed{10}

The chance of getting any given number is

\dfrac{1}{n}

, so the probability of getting

7

and heads is either

\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}

\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}

. We get that either

n = 5

n = 10

, but since rolling a

7

is possible, only

n = \boxed{10}

is a solution.

Problem Statement