Question

The probability of a coin showing head is $p$ and then 100 such coins are tossed. If the probability of 50 coins showing head is same as the probability of 51 coins showing head, then $p$ equals

• A. $\frac{1}{2}$
• B. $\frac{49}{100}$
• C. $\frac{51}{101}$
• D. $\frac{50}{101}$

Answer 1

Answer: (C)

It is given that probability of showing head is $p$, then probability of showing tail will be $(1-p)$. Also, probability of showing 50 and 51 heads in tossing a coin 100 times is same. $\therefore P(X=50)=P(X=51)$
$\Rightarrow { }^{100} C_{50}(p)^{50}(1-p)^{50}={ }^{100} C_{51}(p)^{51}(1-p)^{49}$
$\Rightarrow \frac{100 !}{50 ! 50 !}(1-p)=\frac{100 !}{51 ! 49 !} p$
$\Rightarrow \frac{1-p}{50}=\frac{p}{51}$
$\Rightarrow 51-51 p=50\,p$
$\Rightarrow p=\frac{51}{101}$