In eight throws of a die, 5 or 6 is considered a success. The mean and standard deviation of total number of successes is respectively given by

Concept

When a fair dice is rolled, probability of getting any number is \(\frac{1}{6}\).

Suppose an experiment is repeated n times. Probability of success be p.

If X denote the number of success, then

P(X = k) = C(n, k) × p^k (1 – p)^{n - k} , 0 ≤ k ≤ n

where

\({\bf{C}}\left( {{\bf{n}},\;{\bf{k}}} \right) = \;\frac{{{\bf{n}}!}}{{{\bf{k}}!\left( {{\bf{n}} - {\bf{k}}} \right)!}}\)

Mean of X = np

Standard deviation of X \(= \sqrt {{\bf{np}}\left( {1 - {\bf{p}}} \right)}\)

Calculation

A dice is rolled 8 times so, n = 8.

In a single roll of dice, probability of getting 5 or 6 is

= probability of getting 5 + probability of getting 6

\(= \frac{1}{6} + \frac{1}{6}\)

\(= \frac{1}{3}\)

So probability of success \({\bf{p}} = \frac{1}{3}\)

\({\bf{Mean}} = {\bf{n}} \times {\bf{p}} = \frac{8}{3}\)

\({\bf{Standard}}\;{\bf{deviation}} = \sqrt {{\bf{np}}\left( {1 - {\bf{p}}} \right)} = \frac{4}{3}\)

Correct option is (2).