Question

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

 

Answer 1

Let X be the number of heads that appear when a coin is tossed 5 times.
X follows a binomial distribution with n =5 \[\text{ and }  p = q = \frac{1}{2}\]

\[P(X = r) = ^{5}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{5 - r} \]
\[ =^{5}{}{C}_r \left( \frac{1}{2} \right)^5 \]

\[P (\text{ head appears an even number of  times }) = P(X = 0) + P(X = 2) + P(X = 4)\]
\[ = ^{5}{}{C}_0 \left( \frac{1}{2} \right)^5 + ^{5}{}{C}_2 \left( \frac{1}{2} \right)^5 + ^{5}{}{C}_4 \left( \frac{1}{2} \right)^5 \]
\[ = \frac{1 + 10 + 5}{2^5}\]
\[ = \frac{16}{32}\]
\[ = \frac{1}{2}\]