Question

Let X denote the number of times heads occur in n tosses of a fair coin. If P(X = 4), P (X = 5) and P(X = 6) are in A.P.; then

value of n is -

• A. 7, 14
• B. 10
• C. 12
• D. None of these

Answer 1

Answer: (A)

7, 14

Answer 2

Clearly, X is a binomial variate with parameters n and p

= $\frac { 1 } { 2 }$ such that

P (X = r) = ⁿC_r p^r q^n–r = ⁿC_r $\left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } - \mathrm { r } }$= ⁿC_r $\left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } }$

Now, P (X = 4), P (X = 5) and P (X = 6) are in A.P.

Ž 2P (X = 5) = P (X = 4) + P (X = 6)

Ž 2 . ⁿC₅ $\left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } }$ = ⁿC₄ $\left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } }$ + ⁿC₆ $\left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } }$ Ž 2 ⁿC₅ = ⁿC₄ + ⁿC₆

Ž 2 $\frac { n ! } { ( n - 5 ) ! 5 ! }$ = +Ž $\frac { 2 } { 5 ( n - 5 ) }$

= $\frac { 1 } { ( n - 4 ) ( n - 5 ) }$+ $\frac { 1 } { 6 \times 5 }$

Ž n² – 21n + 98 = 0 Ž (n – 7) (n – 14) = 0 Ž n = 7 or 14.

Hence (1) is correct answer