Question

A and B are Involved in a duel. The rules of the duel are that they are to shoot each other simultaneously. If one or both are hit, then the duel is over. If shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of A will hit B with probability 0.5, and each shot of B will hit A with probability 0.4

1. What is the probability that the duel ends at the first round?

2. Wha is the probability that the duel ends at the 4th round of shots?

3. Given that the duel ends at the 4th round, what is the probability that A is not hit?

Answer 1

0 Given that, probability o each shot of A will hit B with 0.5 (let us call it 'a') and each shot of B will hit A with probability 0.41 let us call it b) So, a=0.5 and b=0.4 to let us consider that the dual ends on 'mith "trial, so this means no one gets hit on the first m-1'the trials and then some one gets hit on m'th trial; this can happen when (A gets hit, B doesn't get hit), (A doesn't get hit, B gets hit) and (A gets hit, gets hit) and so the probability of some one getting hit is not a (1-6) + (1-a) bt ab since = a-abt b-abtab they are g. = atb-ab independent = "-( I-a- btab) =1-1 -a)-bl1-a) = -(1-2) (1-6). and, the probability of no one getting hit on the first map trials is (1-ajm-1 (1-6) m-1 since they are independent. Thus, the berobability that the dual ends on exact 'm'th trial is [1-(1-2) (1-2)] (1-a) m-1 (1-6) m- (as they are independent).

the 1.) m =1, So, the brobability th dual ends at the first round is a 1- ( l-a) (1-6) tes - 1-(1-0.5) (1-0-4) D = 1- 0.5 80-6 = 1-0.3= 0.7 2.) m-4, So, the probability that the dual ends at the 4th nound is [1-(1-2) (1-2)] (1-2) 4-1 (1-64- 1 0 = 0.7 X (0.5) 3 x (0.6)3 0.7 X 0.027 = 0.0189. ) 3. Let us call x be the event that A is not hit, and y be the event that the dual ends on 4th nound. So, we need to find P(xly) i.e. Pr A is not hit the duala ends on the 4th bound) Now, p(xny) = P(A is not hit and the et dual ends on the traut 4th nound)

= a (1-6) (1-9) 4-4(1-1) 14 I at last in A is not hit, but - Bishit ? And P(y) = [1- (1-2) (1-2)] (1-2) 4-1 (1-6)Ay So, p(x(Y) = P(xny) (by ) - [1-(1-4) (1-5)]((1-673 0.5 x 0-6 [ - 0.550.61 0.7 So, the Given that the duel ends at the 4th sound, the probability that A is hit