Question

One hundred identical coins, each with probability p of showing heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, the value of p is
(a) 1/2
(b) 51/101
(c) 49/101
(d) None of these

Answer 1

(b) 51/101

$$\begin{gathered} \mathrm{Let}\mathit{}X\mathrm{denote}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{coins}\mathrm{showing}\mathrm{head}. \\ \mathrm{Therefore},X\mathrm{follow}s\mathrm{a}\mathrm{binomial}\mathrm{distribution}\mathrm{with}p\mathrm{and}n\mathrm{as}\mathrm{parameters}. \\ \mathrm{Given}\mathrm{that}P(X=50)=P(X=51) \\ \Rightarrow {}^{100}C_{50}{p}^{50}{q}^{50}={}^{100}C_{51}{p}^{51}{q}^{49} \\ \mathrm{on}\mathrm{simplifying}\mathrm{we}\mathrm{get}, \\ \frac{51}{50}=\frac{p}{q} \\ \Rightarrow \frac{51}{50}=\frac{p}{1-p}(\sin cep+q=1) \\ \Rightarrow p=\frac{51}{101} \end{gathered}$$