Question

2. The probabilities with which we choose one of three target shooters are 0.2 for Angela, 0.5 for Brunhilde, and 0.3 for Chelsea. The probabilities of each hitting a target from one shot are 0.3 for Angela, 0.4 for Brunhilde, and 0.1 for Chelsea.

(a) The chosen shooter fires one shot and misses the target. Find the probability that we have chosen Angela.

(b) Are the events that Angela was chosen and the target was missed positively related, negatively related, or independent? Justify your answer.

(c) Suppose Angela and Brunhilde each fired one shot at the target. If one bullet is found in the target, find the probability that it came from Brunhilde’s gun. Assume that the two shooters fire at the target independently.

Answer 1

Let X denote the event that the bullet shot hits the target.

A denote the event that the target shooter is Angela

B denote the event that the target shooter is Brunhilde

C denote the event that the target shooter is Chelsea.

From given data,

P(A) = 0.2, P(B) = 0.5, P(C) = 0.3

P(X|A) = 0.3, P(X|B) = 0.4, P(X|C) = 0.1

(a) Now,

P(X) = P(A)*P(X|A) + P(B)*P(X|B) + P(C)*P(X|C)

= 0.2*0.3 + 0.5*0.4 + 0.3*0.1 = 0.29

Thus, probability that bullet shot do not hits the target, P(Cc) = 1 - P(X) = 0.71

Now, probability that we have chosen Angela when the chosen shooter fires one shot and misses the target = P(A|Xc)

= P(Xc|A)*P(A)/P(Xc)

Now, P(Xc|A) = 1 - P(X|A) = 0.7

Thus, P(A|Xc) = 0.7*0.2/0.71 = 0.197

(b) To find the dependence between A and Xc

Now, P(Xc|A) = 0.7 and P(Xc) = 0.71

Since, both the probabilities are not equal, they are dependent.

Now, it can be seen that P(Xc|A) < P(Xc) and P(A|Xc) < P(A). This implies that the probability of the given events decreases with the occurrence of the second event than it was independently.

Thus, the events are negatively related.

(c) Let M be the event that one out of the two bullet fired one each by Angela and Brunhilde hits the target.

And N be the event that the bullet shot by Brunhilde hits the target and the one shot by Angela does not hit the target.

P(N) = 0.4*(1 - 0.3) = 0.28

Now, P(M) = 0.4*(1 - 0.3) + 0.3*(1 - 0.4) = 0.46

Thus, required probability = 0.28/0.46 = 0.609