Problem 133

Solution 1:

Suppose that the six-sided die is rolled first. The result of this
roll is a positive integer from $1$
to $6$ inclusive. If the eight-sided
die is rolled second, the probability that the number rolled is the same
as the first number rolled is $\dfrac18$.

Solution 2:

Suppose that the eight-sided die is rolled first. The result of this
roll is a positive integer from $1$
to $8$ inclusive. The probability
that the first roll is a number from $1$ to $6$ inclusive is $\dfrac68$. (If a $7$ or $8$ is rolled on the first die, the
probability of rolling the same number on both dice is $0$.) If the six-sided die is rolled
second, the number rolled is the same as the first roll if the first
roll is a number from $1$ to $6$ inclusive, which occurs with
probability $\dfrac68$, and the
second roll matches the first, which occurs with probability $\dfrac16$. Thus the probability that the
same number is rolled on both dice is $\dfrac68\times\dfrac16=\dfrac18$.

Solution 3:

Consider the ordered pairs $(a,b)$
for which $a$ is the result of
rolling the six-sided die and $b$ is
the result of rolling the eight-sided die.

There are $6$ possible values of
$a$ and $8$ possible values of $b$, and thus there are $6\times8=48$ ordered pairs representing
the results of rolling the two dice.

Of these $48$ ordered pairs, there
are $6$ for which the same number is
rolled on both dice. These are $(1,1)$, $(2,2)$, $(3,3)$, $(4,4)$, $(5,5)$, and $(6,6)$, and so the probability that the
same number is rolled on both dice is $\dfrac{6}{48}=\dfrac18$.