A and also B are Involved in a duel. The rules of the duel are thatthey are to shoot each other at the same time. If one or both arehit, then the duel is over. If shots miss out on, then they repeat theprocess. Suppose that the results of the shots are independent andthat each shot of A will hit B through probcapability 0.5, and also each shotof B will hit A through probcapacity 0.4

1. What is the probcapability that the duel ends at the firstround?

2. Wha is the probcapacity that the duel ends at the 4th round ofshots?

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

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0 Given that, probcapability o each swarm of A will hit B via 0.5 (let us contact it 'a') and also each shot of B will hit A via probcapability 0.41 let us speak to it b) So, a=0.5 and also b=0.4 to let us consider that the dual ends on 'mith "trial, so this suggests no one gets hit on the first m-1'the trials and also then some one gets hit on m'th trial; this deserve to take place as soon as (A gets hit, B doesn't acquire hit), (A doesn't obtain hit, B gets hit) and also (A gets hit, gets hit) and also so the probability of some one acquiring hit is not a (1-6) + (1-a) bt ab considering that = a-abt b-abtab they are g. = atb-ab independent = "-( I-a- btab) =1-1 -a)-bl1-a) = -(1-2) (1-6). and also, the probability of no one obtaining hit on the initially map trials is (1-ajm-1 (1-6) m-1 considering that they are independent. Thus, the berobcapacity that the dual ends on precise '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 brobcapacity 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 probcapacity 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 contact x be the event that A is not hit, and y be the event that the dual ends on 4th nound. So, we have to uncover 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 also the et dual ends on the traut fourth nound)

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= a (1-6) (1-9) 4-4(1-1) 14 I at last in A is not hit, however - 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