Question

Probability of a man hitting a target is 1/4, then the number of times a man must fire to hit a target so that probability of his hitting the target at least once is greater than 2/3 is 

(A) 2

(B) 4

(C) 6

(D) none of these

Answer 1

Correct option is (B) 4

Suppose the man fires n times and let X denote the number of times he hits the target. Then,

\(P(X = r) = nC_r\left(\frac 14\right)^r \left(\frac 34\right)^{n - r}\)

r = 0, 1, 2,.., n

It is given that

\(P(X \ge 1) > \frac 23\)

\(1 - P(X = 0) > \frac 23\)

\(1 - n_{C_0} \left(\frac 14\right)^0\left(\frac 34\right)^n > \frac 23\)

\(1-\left(\frac 34\right)^n > \frac 23\)

\(\left(\frac 34\right)^n < \frac 13\)

⇒ n = 4, 5, 6,....

Hence, the man must fire at least 4 times.