Question

Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.

Answer 1

Total number of outcomes when two dice are thrown = 6 x 6, i.e. 36
Let X be the number of doublets in three throws of a pair of dice.
Then, X follows a binomial distribution with n = 3

$$p=P(\text{ Getting a doublet in three throws })=\frac{6}{36}=\frac{1}{6}\text{ and }q=\frac{5}{6}$$
$$\therefore P(X=r){=}^3{C}_r{\left(\frac{1}{6}\right)}^r{\left(\frac{5}{6}\right)}^{3-r},r=0,1,2,3$$
$$\text{ Mean }\left(\text{ np }\right)=3\left(\frac{1}{6}\right)=\frac{1}{2}$$

$$\text{ The distribution is as follows }:$$
   X        0             1                 2                 3

$$P(X){\left(\frac{5}{6}\right)}^{3}3{\left(\frac{1}{6}\right)}^1{\left(\frac{5}{6}\right)}^{3-1}3{\left(\frac{1}{6}\right)}^2{\left(\frac{5}{6}\right)}^{3-2}{\left(\frac{1}{6}\right)}^3$$
$$\frac{125}{216}\frac{75}{216}\frac{15}{216}\frac{1}{216}$$