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प्रश्न
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
विकल्प
1/2
51/101
49/101
None of these
उत्तर
51/101
\[\text{ Let X denote the number of coins showing head .} \]
\[\text{ Therefore, X follows a binomial distribution with p and n as parameters . } \]
\[\text{ Given that } P(X = 50) = P(X = 51)\]
\[ \Rightarrow ^{100}{}{C}_{50} \ p^{50} q^{50} = ^{100}{}{C}_{51} \ p^{51}\ q^{49} \]
\[\text{ on simplifying we get } , \]
\[\frac{51}{50} = \frac{p}{q}\]
\[ \Rightarrow \frac{51}{50} = \frac{p}{1 - p} (\text{ since} \ p + q = 1)\]
\[ \Rightarrow p = \frac{51}{101}\]
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