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# 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 - Mathematics

ConceptBernoulli Trials and Binomial Distribution

#### 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

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

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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.
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