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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.
Solution
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
\[P(X) \ \frac{64}{343} \frac{144}{343} \frac{108}{343} \frac{27}{343}\]
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