Question

Let X denote the number of times heads occur in n tosses of a fair coin. If P(X=4),P(X=5)andP(X=6) are in AP; the value of n is 7,14 b. 10,14 c. 12,7 d. 14,12 A. 7,14 B. 10,14 C. 12,7 D. 14,12 Select the correct answer from above options

Answer 1

Correct Answer - A Clearly, X is a binomial variate with parameters n and p=1/2 such that P(X=r)=.nCrprqn-1=.nCr( 1 2 )r( 1 2 )n-r=.nCr( 1 2 )r ltNow, P(X=4),P(X=5)andP(X=6) are in A.P. ⇒2P(X=5)=P(X=4)+P(X=6) 2.nC5( 1 2 )n=.nC4( 1 2 )n+.nC6( 1 2 )n ⇒2.nC5=.C_4+.nC6 ⇒2 ǃ (n-5)ǃ5ǃ = nǃ (n-4)ǃ4ǃ + nǃ (n-6)ǃ6ǃ ⇒ 2 5(n-5) = 1 (n-4)(n-5) + nǃ 6x5 ⇒n2-21n+98=0⇒(n-7)(n-14)=0⇒n=7or14 .