Question

A die is rolled four times. If the probability that product of first 3 outcomes is equal to fourth outcome is p, then $\left[\frac{1}{\sqrt{p}}\right]$ is equal to (where [.] denotes greatest integer function) ___.

Answer 1

$p={\left(\frac{1}{6}\right)}^3\left(\frac{1}{6}\right)+3{\left(\frac{1}{6}\right)}^3\left(\frac{1}{6}\right)+3{\left(\frac{1}{6}\right)}^3\left(\frac{1}{6}\right)+6{\left(\frac{1}{6}\right)}^3\left(\frac{1}{6}\right)+3{\left(\frac{1}{6}\right)}^3\left(\frac{1}{6}\right)+9{\left(\frac{1}{6}\right)}^3\left(\frac{1}{6}\right)=\frac{25}{6^4}$

$\Rightarrow \left[\frac{1}{\sqrt{p}}\right]=7$