A king and his army will attempt to capture a fortress. The left and right flank

Question

A king and his army will attempt to capture a fortress. The left and right flanks break off from the main group to attack the west and east guard towers. Suppose the left flank has a 60% chance of success and the right flank has a 75% chance of success, independently of one another. If both flanks capture their respective targets, then the king has a 98% chance of successfully taking the fort. If, however, only the left flank captures its tower, the king has an 80% chance of success; if only the right flank succeeds, the king has a 50% chance. If both flanks fail, then the king’s chance of capturing the fort drops to 20%. It turns out the king captures the fort. What is the probability that one, and only one, flank was successful (either the left, or the right, but not both)

Explanation / Answer

P(only left flank capture target) =0.6-0.6*0.75=0.15

P(only right flank capture target) =0.75-0.6*0.75=0.3

P(both flank capture target) =0.6*0.75=0.45

P(both flank fail) =1-(0.6+0.75-0.45)=0.1

hence P(king captures the fort) =P(only left succeed and king capture+only right succeed and king capture+both succeed and king capture+both fail and king capture)

=0.15*0.8+0.3*0.5+0.45*0.98+0.1*0.2=0.731

hence   probability that one, and only one, flank was successful given king captures the fort

=P(only left succeed and king capture+only right succeed and king capture)/P(king captures the fort)

=(0.15*0.8+0.3*0.5)/0.731=0.3694

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