#### Question

A fair coin is tossed 99 times. If *X* is the number of times head appears, then *P* (*X* = *r*) is maximum when *r* is

49, 50

50, 51

51, 52

None of these

#### Solution

49, 50

When a coin is tossed 99 times, the number of heads *X* follows a binomial distribution with

\[p = q = \frac{1}{2} = 0 . 5\]

\[P(X = r) = ^{n}{}{C}_r (0 . 5 )^r (0 . 5 )^{n - r} = ^{n}{}{C}_r (0 . 5 )^n \]

\[As (0 . 5 )^n \text{ is common to all r it is enough if we find the maximum of }\ ^{\ n}{}{C}_r . \]

\[\text{ We know that for odd number of n, there will be two equal maximum terms, } \]

\[\text{ i . e . when } r = \frac{n - 1}{2}\text{ and } r = \frac{n + 1}{2}\]

\[\text{ Hence,} \ n = 99 \]

\[\text{ So, the maximum is obtained when r = 49 or } 50\]

Is there an error in this question or solution?

Solution A Fair Coin is Tossed 99 Times. If X is the Number of Times Head Appears, Then P (X = R) is Maximum When R is (A) 49, 50 (B) 50, 51 (C) 51, 52 (D) None of These Concept: Bernoulli Trials and Binomial Distribution.