Question

An urn contains four white and three red balls. Find the probability distribution of the number of red balls in three draws with replacement from the urn.

Answer 1

As three balls are drawn with replacement, the number of white balls, say X, follows binomial distribution with n =3

\[p = \frac{3}{7} \text{ and } q = \frac{4}{7}\]
\[P(X = r) = ^{3}{}{C}_r \left( \frac{3}{7} \right)^r \left( \frac{4}{7} \right)^{3 - r} , r = 0, 1, 2, 3\]
\[P(X = 0) = ^{3}{}{C}_0 \left( \frac{3}{7} \right)^0 \left( \frac{4}{7} \right)^{3 - 0} \]
\[ P(X = 1) = ^{3}{}{C}_1 \left( \frac{3}{7} \right)^1 \left( \frac{4}{7} \right)^{3 - 1} \]
\[P(X = 2) = ^{3}{}{C}_2 \left( \frac{3}{7} \right)^2 \left( \frac{4}{7} \right)^{3 - 2} \]
\[P(X = 3) = ^{3}{}{C}_3 \left( \frac{3}{7} \right)^3 \left( \frac{4}{7} \right)^{3 - 3} \]
  X         0     1      2      3
\[P(X) \                        \frac{64}{343} \frac{144}{343} \frac{108}{343} \frac{27}{343}\]