Question

Find the probability distribution of the number of sixes in three tosses of a die.

 

Answer 1

Let X be the number of 6 in 3 tosses of a die.
Then X follows a binomial distribution with n =3.

\[p = \frac{1}{6}, q = 1 - p = \frac{5}{6}\]
\[P(X = r) = ^{3}{}{C}_r \left( \frac{1}{6} \right)^r \left( \frac{5}{6} \right)^{3 - r} , r = 0, 1, 2, 3\]
\[P(X = 0) = ^{3}{}{C}_0 \left( \frac{1}{6} \right)^0 \left( \frac{5}{6} \right)^{3 - 0} \]
\[P(X = 1) =^{3}{}{C}_1 \left( \frac{1}{6} \right)^1 \left( \frac{5}{6} \right)^{3 - 1} \]
\[P(X = 2) = ^{3}{}{C}_2 \left( \frac{1}{6} \right)^2 \left( \frac{5}{6} \right)^{3 - 2} \]
\[P(X = 3) = ^{3}{}{C}_3 \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^{3 - 3} \]

\[\text{ Hence, the distribution of X is as follows }  . \]
   X       0      1      2       3
\[P(X) \       \             \frac{125}{216} \frac{75}{216} \frac{15}{216} \frac{1}{216}\]