Question

Suppose that a trainee soldier shoots a target in an independent manner. If the probability that the

target is hit on any one shot is 0.8, what is the probability that the target would be hit

i. on the sixth attempt

ii. in fewer than 5 shots

iii. in even number of shots

Answer 1

The Geometric distribution is

$$P(X=x)=p(x)= q^{x-1}p,x= 1, 2, 3, ...$$

Given $p=0.8, q=1-p=0.2$

i.

$$P(X=6)=(0.2)^{6-1}(0.8)=0.000256$$

ii.

$$P(X<5)=P(X=1)+P(X=2)+P(X=3)$$

$$+P(X=4)=(0.2)^{1-1}(0.8)+(0.2)^{2-1}(0.8)$$

$$+(0.2)^{3-1}(0.8)+(0.2)^{4-1}(0.8)=0.9984$$

iii.

$$P(X\ is \ even\ number)=P(X=2)+P(X=4)$$

$$+P(X=6)+...=0.8(0.2)^{2-1}+0.8(0.2)^{4-1}$$

$$+0.8(0.2)^{6-1}=0.8(0.2+(0.2)^3+(0.2)^5+...)$$

$$=0.8(0.2)(\dfrac{1}{1-(0.2)^2})=\dfrac{1}{6}$$