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

Answer 1

Explanation:

Let the probability of winning from left side be L and from right side by R .

Let the probability of winning by king be K .

Given information can be summarised as follows :

P(L) = .6

P(R) = .75

P K/(L∩R)=.98

P K/(L∩R')= .8

P K/(L'∩R) = .5

P K/(L∩R)' = .2

Probability of win by King can be summarised as follows .

P(k) = P(L∩R) x P K/(L∩R) + P(L∩R') x P K/(L∩R') + P(L'∩R) x P K/(L'∩R) + P(L∪R)' x P K/(L∪R)'

P(L∩R) = .6 X .75 = .45

P(L∪R) = P(L) +P(R) - P(L∩R)

= .6 +.75 - .45 = .90

P(L∪R)' = 1 - P(L∪R) = .10

P(L∩R') = P(L) - P(L∩R) = .6 - .45 = .15

P(L'∩R) = P(R) - P(L∩R) = .75 - .45 = .30

P(k) = P(L∩R) x P K/(L∩R) + P(L∩R') x P K/(L∩R') + P(L'∩R) x P K/(L'∩R) + P(L∪R)' x P K/(L∪R)'

= .45 x .98 + .15 x .8 + .30 x .5 + .1 x .2

= .441 + .12 + .15 + .02

= .73 approx .