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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

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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

  • 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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