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Problem of the Week
Problem D and Solution
Not So Random

Problem

Kimi created a digital die that can be controlled with a program. She then programmed it as follows.

Kimi rolls the die once and the numbers on the die change as described above. She then rolls the die again, but this time something goes wrong and none of the numbers change. What is the probability that she rolled a \(2\) on her second roll?

Solution

Solution 1

In this solution, we will determine the possibilities for the first and second roll to count the total number of possible outcomes. We will then count the number of outcomes in which the second roll is a \(2\) and determine the probability.

There are \(36\) possible outcomes in total. Of these outcomes, \(8\) have a second roll of \(2\). Therefore, the probability of rolling a \(2\) on the second roll is \(\frac{8}{36}=\frac{2}{9}\).

Solution 2

In this solution, we will show the possibilities on a tree diagram.

To calculate the probability of rolling an odd number on the first roll and then a \(2\) on the second roll, we multiply the probabilities of each to obtain \(\frac{1}{3}\times \frac{1}{3}=\frac{1}{9}\).

To calculate the probability of rolling an even number on the first roll and then a \(2\) on the second roll, we multiply the probabilities of each to obtain \(\frac{2}{3}\times \frac{1}{6}=\frac{1}{9}\).

Then, to calculate the probability of rolling an odd number on the first roll and then a \(2\) on the second roll, or an even number on the first roll and then a \(2\) on the second roll, we add their probabilities to obtain \(\frac{1}{9}+\frac{1}{9}=\frac{2}{9}\).

Therefore, the probability of rolling a \(2\) on the second roll is \(\frac{2}{9}\).