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

$\frac{51}{101}$

We have ${^{100}C}_{50}p^{50} (1-p)^{50}= {^{100}C}_{51}p^{51} (1-p)^{49}$  or      $ \frac{1-p}{p}=\frac{100!}{51!.49!}×\frac{50!.50!}{100!}=\frac{50}{50}$  or

$51 - 51 p = 50p ⇒ = \frac{51}{101}.$