Question

A coin is tossed 5 times. What is the probability that tail appears an odd number of times?

 

Answer 1

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