Question

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

Answer 1

$P\left(minimumfacevalueisnotlessthan2andmaximumfacevalueisnotgreaterthan5\right)$

$=P\left(2or3or4or5\right)=\frac{4}{6}=\frac{2}{3}$

Hence required probability ${=}^4{C}_4{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^0=1\times \frac{16}{81}\times 1=\frac{16}{81}$