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Question
Find the probability of 4 turning up at least once in two tosses of a fair die.
Solution
Let X be the probability of getting 4 in two tosses of a fair die.
X follows a binomial distribution with n =2; \[p = \frac{1}{6} \text{ and } q = \frac{5}{6};\]
\[P(X = r) = ^{2}{}{C}_r \left( \frac{1}{6} \right)^r \left( \frac{5}{6} \right)^{2 - r} \]
\[\text{ Probability of getting 4 at least once } = P(X \geq 1) \]
\[ = 1 - P(X = 0) \]
\[ = 1 - ^{2}{}{C}_0 \left( \frac{1}{6} \right)^0 \left( \frac{5}{6} \right)^{2 - 0} \]
\[ = 1 - \frac{25}{36}\]
\[ = \frac{11}{36}\]
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