The probability of a man hitting a target is 2 3 , the minimum number of times he must fire so that the probability of hitting the target at least once is more then 90 %

Step-by-step solution

Let X the number of times, he hits the targets. Hitting the target is a Bernoulli trial, So, X has a binomial distribution. P(X=x)= ^n C_x q^ n-x p^x where, n = Number of times hit p = probability of hitting = 2/ 3 q = probability of not hitting = 1 − 2/ 3 = 1 / 3 P(X=x)= ^n C_x ( 1 3 )^ n-x ( 2 3 )^x Given, P(x 1)>90 % aligned & 1-P(X=0)>90 % & 1- ^n C₀ ( 1 3 )^n ( 2 3 )^0>0.9 & 1- ( 1 3 )^n>0.9 0.1> ( 1 3 )^n & 3^n>10 n 3 aligned Minimum number of times to hit the target =3