Question

An unbiased die with faces marked $1,2,3,4,5$ and $6$ is rolled four times. Out of four values obtained the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

• A. $\frac{16}{81}$
• B. $\frac{1}{81}$
• C. $\frac{80}{81}$
• D. $\frac{65}{81}$

Answer 1

The correct option is A

$\frac{16}{81}$

Explanation for the correct option:

Step: 1 Find out the probability that the face values are from $2$ to $5$.

The minimum face value is not less than $2$and the maximum face value is not greater than $5$.

Therefore, the number of possible face values are $4$.

Therefore, the probability that the minimum face value is not less than $2$and the maximum face value is not greater than $5$ is

$$\begin{gathered} =\frac{4}{6} \\ =\frac{2}{3} \end{gathered}$$

Step: 2 Find the required probability.

Therefore, the required probability

$$\begin{gathered} =C_4^4{\left(\frac{2}{3}\right)}^4{\left(\frac{2}{3}\right)}^{4-4}={\left(\frac{2}{3}\right)}^4 \\ =\frac{16}{81} \end{gathered}$$

Therefore, Option(A) is the correct answer.