Question

Question: What is the probability that the sum of the numbers on two dice is even when they are rolled?

Short answer

Answer

The probability that the sum of the numbers on two dice is even when they are rolled is$P\left(E\right)=0.5$$P\left(E\right)=0.5$.

Step-by-step solution

 Given  

The sum of the numbers on two dice is even when they are rolled.

The Concept of Probability

Ifrepresents the sample space and$\mathit{E}$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$\mathbf{}\mathit{P}\mathbf{\left(}\mathit{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{n}\mathbf{\left(}\mathbf{E}\mathbf{\right)}}{\mathbf{n}\mathbf{\left(}\mathbf{S}\mathbf{\right)}}$

Determine the probability

As per the problem we have been asked to find that the sum of the numbers on two dice is even when they are rolled.

If represents the sample space and $E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$

When two dice are thrown the all the possible combinations are follows:

$\left\{\begin{array}{c}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)\\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6)\\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6)\\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$

So total number of events in sample space can be given as $n\left(s\right)=36$

Now all the possible cases when the sum of the numbers on two dice are even are as follows:

$\left\{\begin{array}{c}(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5)\\ (4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4),(6,6)\end{array}\right\}$

So, the number of favourable events $n\left(E\right)=18$.

Now substitute the values in the above formula we get

$$\begin{gathered} P\left(E\right)=\frac{18}{36} \\ P\left(E\right)=\frac{1}{2} \\ P\left(E\right)=0.5 \end{gathered}$$

The probability that the sum of the numbers on two dice is even when they are rolled is$P\left(E\right)=0.5$.