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(exactly one flank was successful | king captures the fort) = P(exactly one flank was successful and king captured the fort) / P(king captured the fort)

= (0.6x0.25x0.8 + 0.4x0.75x0.5)/(0.4x0.25x0.20 + 0.6x0.25x0.8 + 0.4x0.75x0.5 + 0.6x0.75x0.98)

= 0.27/0.731

= 0.3694

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