Question

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is
(a) 1/2
(b) 1/8
(c) 3/8
(d) None of these

Answer 1

(a) 1/2

Here n=100
Let X denote the number of times a tail is obtained.

$$\begin{gathered} \mathrm{Here},\mathit{}p=q=\frac{1}{2} \\ P(X=\mathrm{odd})=P(X=1,3,5,....99) \\ =\left({}^{100}\mathrm{C}_1+{}^{100}\mathrm{C}_3+.....+{}^{100}\mathrm{C}_{99}\right){\left(\frac{1}{2}\right)}^{100} \\ =\mathrm{Sum}\mathrm{of}\mathrm{odd}\mathrm{coefficient}\text{s}\mathrm{in}\mathrm{binomial}\mathrm{expansion}\mathrm{in}(1+x{)}^{100}{\left(\frac{1}{2}\right)}^{100} \\ =\frac{2^{(100-1)}}{2^{100}} \\ =\frac{1}{2} \end{gathered}$$