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 (a) on the sixth attempt (b) in fewer than 5 shots () in even number of shots. Ans.(a) 0.00026; (b) 0.9984; (c) 0.166

Answer 1

Given: Probability (p)=0.8

           Q=1-p=0.2

To Find: the probability that the target would be hit

(a) on the sixth attempt

(b) in fewer than 5 shots

(c) in even number of shots

Solution: The Geometric Distribution is

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

Here we have Probability (p)=0.8

                                      Q=1-p=0.2

• (a)    

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

• (b)

             $P(X<5)=P(X=1)+P(X=2)+P(X=3)+P(X=4)\\ \ =(0.2)^{1-1} (0.8)+(0.2)^{2-1}(0.8)+ (0.2)^{3-1} (0.8)+(0.2)^{4-1}(0.8)\\=0.9984$

• (c)

             $P(X is \ even\ number)=P(X=2)+P(X=4)+P(X=6)+...\\=(0.2)^{2-1} (0.8)+(0.2)^{4-1}(0.8)+ (0.2)^{6-1}\\=0.8(0.2+(0.2)^{3} +(0.2)^{5} +...)\\=0.8(0.2)(\frac{1}{1-(0.2)^{2} } )=\frac{1}{6}$

Answer:-(a) 0.00026; (b) 0.9984; (c) 0.166