Question

A bag contains 10 balls, each marked with one of the digits from 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

Answer 1

Let X be the number of balls marked with the digit 0 when 4 balls are drawn successfully with replacement.
As this is with replacement, X follows a binomial distribution with n = 4;

\[p = \text{ probabilty that a ball randomly drawn bears digit } 0 = \frac{1}{10}; q = 1 - p = \frac{9}{10};\]

\[P(X = r) = ^{4}{}{C}_r \left( \frac{1}{10} \right)^r \left( \frac{9}{10} \right)^{4 - r} \]
\[P(\text{ none bears the digit } 0) = P(X = 0)\]
\[ = ^{4}{}{C}_0 \left( \frac{1}{10} \right)^0 \left( \frac{9}{10} \right)^{4 - 0} \]
\[ = \left( \frac{9}{10} \right)^4\]