Library / /202 of 213

, 2026

Combinatorics Difficulty 4.8 AIME Find the answer Canada

A die is created with the integers $1$ , $2$ , $2$ , $4$ , $7$ , $8$
on its six faces. The first time the die is rolled, the numbers on its
faces are changed in the following way:

If the number rolled is odd, then all odd numbers on the die are
doubled, and all even numbers remain the same.
If the number rolled is even, then all even numbers on the die
are halved, and all odd numbers remain the same.

The die is then rolled a second time. What is the probability that
the second number rolled is $2$ ?

Pick one

Solution

The probability that the second roll is a $2$ is equal to the probability that the
first roll is odd and the second roll is a $2$ , added to the probability that the
first roll is even and the second roll is a $2$ .

The original die has $2$ odd numbers
and $4$ even numbers on its faces.
Thus the probability that the first roll is odd is $\dfrac26$ , and the probability that the
first roll is even is $\dfrac46$ .

If the first roll is odd, the numbers on the faces are changed to
$2$ , $2$ , $2$ , $4$ , $14$ , $8$ .

In this case, the probability that the second roll is a $2$ is $\dfrac36$ , and so the probability that the
first roll is odd and the second roll is $2$ is $\dfrac26\times\dfrac36=\dfrac{6}{36}$ .

If the first roll is even, the numbers on the faces are changed to
$1$ , $1$ , $1$ , $2$ , $7$ , $4$ .

In this case, the probability that the second roll is a $2$ is $\dfrac16$ , and so the probability that the
first roll is even and the second roll is $2$ is $\dfrac46\times\dfrac16=\dfrac{4}{36}$ .

Thus, the probability that the second roll is a $2$ is equal to $\dfrac{6}{36}+\dfrac{4}{36}=\dfrac{10}{36}=\dfrac{5}{18}$ .

Want a route through all this instead of an archive? The track puts 2,444 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.