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 $\mathrm{and}p=q=\frac{1}{2}$
$$\begin{gathered} P(X=r)={}^{5}C_r{\left(\frac{1}{2}\right)}^r{\left(\frac{1}{2}\right)}^{5-r} \\ ={}^{5}C_r{\left(\frac{1}{2}\right)}^5 \end{gathered}$$

$$\begin{gathered} P(\mathrm{head}\mathrm{appear}\text{s an even number of times})=P(X=0)+P(X=2)+P(X=4) \\ ={}^{5}C_0{\left(\frac{1}{2}\right)}^5+{}^{\mathit{5}}C_{\mathit{2}}{\left(\frac{1}{2}\right)}^5+{}^{5}C_4{\left(\frac{1}{2}\right)}^5 \\ =\frac{1+10+5}{2^5} \\ =\frac{16}{32} \\ =\frac{1}{2} \end{gathered}$$