Question

A novice marksman is shooting at a target 1000 feet away. The probability of his hitting the target is 0.3. Your answer should be rounded to the nearest thousandth.

a) What is the probability that he hits the target 3rd time on the 5th shot?

b) What is the probability that the number of shots he will have to make to hit the target is more than 6?

c) What is the probability that he will need 6 shots to hit the target the first time?

d) What is the probability that the number of shots he has to take to hit the target is 10 shots or fewer?

Answer 1

The probability of his hitting the target is 0.3.

a) What is the probability that he hits the target 3rd time on the 5th shot?

Let X be the number of shots required to get 3 hits. We can say that X has a Negative binomial distribution with parameters, number of successes (number of hits) r=3 and success probability ( The probability of his hitting the target) p=0.3

The probability that X=x shots are required to get 3 hits is

the probability that he hits the target 3rd time on the 5th shot is same as the probability that X=5 shots are required to get 3 hits

ans: the probability that he hits the target 3rd time on the 5th shot is 0.079

b) What is the probability that the number of shots he will have to make to hit the target is more than 6?

Let X be the number of shots untill he hits the target. We can sy that X has a Geometric distribution with parameter, success probability (The probability of his hitting the target) p=0.3.

The probaility that X=x shots are required untill he hits the target is

The probability that the number of shots he will have to make to hit the target is more than 6 is same as the probaility that more than 6 shots are required untill he hits the target

ANS: The probability that the number of shots he will have to make to hit the target is more than 6 is 0.118

c) What is the probability that he will need 6 shots to hit the target the first time?

Let X be the number of shots untill he hits the target. We can sy that X has a Geometric distribution with parameter, success probability (The probability of his hitting the target) p=0.3.

The probaility that X=x shots are required untill he hits the target is

the probability that he will need 6 shots to hit the target the first time is

ans: the probability that he will need 6 shots to hit the target the first time is 0.050

d) What is the probability that the number of shots he has to take to hit the target is 10 shots or fewer?

Let X be the number of shots untill he hits the target. We can sy that X has a Geometric distribution with parameter, success probability (The probability of his hitting the target) p=0.3.

The probaility that X=x shots are required untill he hits the target is

the probability that the number of shots he has to take to hit the target is 10 shots or fewer is

ans: the probability that the number of shots he has to take to hit the target is 10 shots or fewer is 0.972