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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
Options
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
\[p = q = \frac{1}{2} = 0 . 5\]
\[P(X = r) = ^{n}{}{C}_r (0 . 5 )^r (0 . 5 )^{n - r} = ^{n}{}{C}_r (0 . 5 )^n \]
\[As (0 . 5 )^n \text{ is common to all r it is enough if we find the maximum of }\ ^{\ n}{}{C}_r . \]
\[\text{ We know that for odd number of n, there will be two equal maximum terms, } \]
\[\text{ i . e . when } r = \frac{n - 1}{2}\text{ and } r = \frac{n + 1}{2}\]
\[\text{ Hence,} \ n = 99 \]
\[\text{ So, the maximum is obtained when r = 49 or } 50\]
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