Question

One hundred identical coins, each with probability p of showing up 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, then the value of p is

• A. $\frac{1}{2}$
• B. $\frac{49}{101}$
• C. $\frac{50}{101}$
• D. $\frac{51}{101}$

Answer 1

The correct option is D $\frac{51}{101}$

Let X be the number of coins showing heads. Let X be a binomial variate with parameters n = 100 and p.
Given that P(X = 50) = P(X = 51)
${\Rightarrow }^{100}{C}_{50}{p}^{50}(1-p{)}^{50}={}^{100}{C}_{51}{p}^{51}(1-p{)}^{49}\Rightarrow \frac{100!}{50!50!}.\frac{51!\times 49!}{100!}=\frac{p}{1-p}\Rightarrow \frac{p}{1-p}=\frac{51}{50}\Rightarrow p=\frac{51}{101}$