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%.

What is the chance the king will capture the fort?  (Give the exact answer.)

If it turns out the king captures the fort, what is the probability that both flanks were successful?  (Give the exact answer.)

If 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)?  (Give the exact answer.)

If it turns out the king fails to capture the fort, what is the probability that neither flank was successful?  (Give the exact answer.)

Answer 1

P(both flanks were successful) = 0.60*0.75 = 0.45

P(only one flank was successful) = 0.60*(1 - 0.75) + (1 - 0.60)*0.75 = 0.45

P(neither flank was successful) = (1 - 0.60)*(1 - 0.75) = 0.10

P(only left flank succeeds) = 0.60*(1 - 0.75) = 0.15

P(only right flank succeeds) = (1 - 0.60)*0.75 = 0.30

The chance that the king will capture the fort = 0.45*0.98 + 0.15*0.80 + 0.30*0.50 + 0.10*0.20 = 0.731

If it turns out that the king captures the fort,

The probability that both flanks were successful

= 0.45*0.98/0.731 = 0.603

The probability that one, and only one, flank was successful = (0.15*0.80 + 0.30*0.50)/0.731 = 0.369

The probability that neither flank was successful = 0.10*0.20/0.731 = 0.027