Chapter 10.1, Problem 6E

Solution

To calculate: The probability of the provided event.

The probability of the events is $\frac{1}{4}$ .

Given information:

A red die and a green die have been rolled.

Need to calculate the probability of events when both numbers are even.

Calculation:

Making a table showing the sums of the two dice helps.

  $\begin{array}{ccccccc}\begin{array}{l}\text{Dice 1 }\to \\ \text{Dice 2}\\ \downarrow \end{array}& 1& 2& 3& 4& 5& 6\\ 1& \left\{\text{1,1}\right\}& \left\{1,2\right\}& \left\{\text{1,3}\right\}& \left\{1,4\right\}& \left\{\text{1,5}\right\}& \left\{1,6\right\}\\ 2& \left\{2,1\right\}& \left\{2,2\right\}& \left\{2,3\right\}& \left\{2,4\right\}& \left\{2,5\right\}& \left\{2,6\right\}\\ 3& \left\{3,1\right\}& \left\{3,2\right\}& \left\{3,3\right\}& \left\{3,4\right\}& \left\{3,5\right\}& \left\{3,6\right\}\\ 4& \left\{4,1\right\}& \left\{4,2\right\}& \left\{4,3\right\}& \left\{4,4\right\}& \left\{4,5\right\}& \left\{4,6\right\}\\ 5& \left\{\text{5,1}\right\}& \left\{5,2\right\}& \left\{5,3\right\}& \left\{5,4\right\}& \left\{5,5\right\}& \left\{5,6\right\}\\ 6& \left\{6,1\right\}& \left\{6,2\right\}& \left\{6,3\right\}& \left\{6,4\right\}& \left\{6,5\right\}& \left\{6,6\right\}\end{array}$

Both numbers are even occurring $9$ times in $36$ possible outcomes.

So, the probability that both numbers are even:

  $\begin{array}{l}P\left(\text{Both numbers are even}\right)=\frac{\text{The number of favorable outcomes}}{\text{Total number of outcomes}}\\ =\frac{9}{36}\\ =\frac{1}{4}\end{array}$

Hence, the probability that both numbers are even is $\frac{1}{4}$ .