Question

Question: What is the probability that a fair die never comes up an even number when it is rolled six times?

Short answer

Answer

The probability that a fair die never comes up an even number when it is rolled six times is$P\left(E\right)=\frac{1}{64}$ .

Step-by-step solution

 Given  

A fair diewhen it is rolled six times.

The Concept of Probability 

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

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

Determine the probability

As per the problem we have been asked to find the probability that a fair die never comes up an even number when it is rolled six times.

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

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

Sample space in rolling a die is$S=\{1,2,3,4,5,6\}$

The favourable event is$E=\{1,3,5\}$

The probability of getting an odd number when a single die is rolled is

$$\begin{gathered} P\left(E\right)=\frac{3}{6} \\ =\frac{1}{2} \end{gathered}$$

As the occurrence of each event is independent so the probability that a fair die never comes up an even number when it is rolled six times can be given as below:

$$\begin{gathered} P\left(E\right)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} \\ P\left(E\right)=\frac{1}{2^6} \\ P\left(E\right)=\frac{1}{64} \end{gathered}$$

The probability that a fair die never comes up an even number when it is rolled six times is $P\left(E\right)=\frac{1}{64}$.