Question

Athos, Porthos and Aramis go target shooting together. Suppose that the chance that Porthos
hits the target is 60%, while Aramis and Athos hit the target with probability 40% and 20%,
respectively.
At the first round, they all shoot at the same target and at the same time. Also, their shots are
independent.
a) Given that exactly one shot hit the target, what is the probability that it was Porthos' shot?
b) Given that the target is hit, what is the probability that Porthos hit it?
On the second round, the shooter is selected randomly.
c) Define the sample space of the second round and draw a tree diagram.
d) Given that the target is hit, what is the probability that the shooter was Athos?

Answer 1

Let A be an event that Atho hits the target, B be an event that Porthos hits the target, C be an event that Aramis hits the target.

According to the problem, P(B)=0.6, P(A)=0.2, P(C)=0.4

a. Let D be an event that exactly one shot hit the target i.e.

b.

Let E be an event that the target is hit.

c. Sample space={ABC, ABC^c, AB^cC, A^cBC, A^cB^cC, A^cBC^c, AB^cC^c, A^cB^cC^c}

where  A is an event that Atho hits the target, B is an event that Porthos hits the target, C is an event that Aramis hits the target, A^c is an event that Atho does not hit the target, B^c is an event that Porthos does not hit the target, C^c is an event that Aramis does not hit the target, ABC is all three hit the target,

ABC^c is Atho, Porthos hit the target and Aramis does not hit the target and so on.

d.