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.
• B.
• C.
• D.

Answer 1

The correct option is D

Let p be the probability of one coin showing head. Then the probability of one coin showing tail is 1 – p. According to question, the coin is tossed 100 times and probability of 50 coins showing head is equal to the probability of 51 coins showing head.
Using binomial probability distribution $P(X=r){=}^n{C}_r{p}^r{q}^{n-r}$,
We get
${}^{100}{C}_{5}0p^{50}(1-p{)}^{50}{=}^{100}{C}_{51}{p}^{51}(1-p{)}^{49}or\frac{1-p}{p}\frac{{}^{100}{C}_{51}}{{}^{100}{C}_{50}}=\frac{50!50!}{51!49!}=\frac{50}{51}or51-51p=50por101p=51orp=\frac{51}{101}$