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Question
A coin is tossed 5 times. What is the probability that tail appears an odd number of times?
Solution
\[\text{ Let X denote the number of tails when a coin is tossed 5 times} . \]
\[\text{ X follows a binomial distribution with n } = 5; p = \frac{1}{2}; q = 1 - p = \frac{1}{2}\]
\[\text{ Then } P(X = r) =^{5}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{n - r} =^{5}{}{C}_r \left( \frac{1}{2} \right)^5 \]
\[\text{ The required probability = P(X = odd)} \]
\[ = P(X = 1) + P(X = 3) + P(X = 5)\]
\[ = ^{5}{}{C}_1 \left( \frac{1}{2} \right)^5 + ^{5}{}{C}_3 \left( \frac{1}{2} \right)^5 +^{5}{}{C}_5 \left( \frac{1}{2} \right)^5 \]
\[ = \left( \frac{1}{2} \right)^5 [5 + 10 + 1] \]
\[ = \frac{16}{32}\]
\[ = \frac{1}{2}\]
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