Question

A fair coin is tossed 99 times. If X is the number of times head appears, then P (X = r) is maximum when r is
(a) 49, 50
(b) 50, 51
(c) 51, 52
(d) None of these

Answer 1

(a) 49, 50

When a coin is tossed 99 times, the number of heads X follows a binomial distribution with
$$\begin{gathered} p=q=\frac{1}{2}=0.5 \\ P(X=r)={}^{n}C_r(0.5{)}^r(0.5{)}^{n-r}={}^{n}C_r(0.5{)}^n \\ \mathrm{As}(0.5{)}^n\mathrm{is}\mathrm{common}\mathrm{to}\mathrm{all}r\mathrm{it}\mathrm{is}\mathrm{enough}\mathrm{if}\mathrm{we}\mathrm{find}\text{the}\mathrm{maximum}\mathrm{of}{}^{n}C_r. \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{for}\mathrm{odd}\mathrm{number}\mathrm{of}n,\mathrm{there}\mathrm{will}\mathrm{be}\mathrm{two}\mathrm{equal}\mathrm{maximum}\mathrm{terms}, \\ \mathrm{i}.\mathrm{e}.\mathrm{when}r=\frac{n-1}{2}\mathrm{and}r=\frac{n+1}{2} \\ \mathrm{Hence},\mathit{}n\mathit{}=99 \\ So,\text{the}\mathrm{maximum}\text{is obtained}\mathrm{when}r=49\mathrm{or}50 \end{gathered}$$