Question

In: Statistics and Probability

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

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)?

Solutions

Expert Solution

L - left flank has been captured

R - right flank has been captured

X - successfully taking the fort

there are 4 cases

(x,y) - x represent win/loss on left flank , y for right

Probability of successfully capturing in each case

1) (W,W)   = 0.6 *0.75 * 0.98

2)(W,L) = 0.6 * (1 - 0.75) * 0.80

3) (L,W) =(1 - 0.6) * 0.75 * 0.5 )

4)(L,L) = 0.4*0.25 * 0.20

we have to find the probability

P(((L-R and R-L)| X)

=( P(W,L) + P(L,W))/P(X)

where P(X) is probability of successfully capturing the fort.

= (0.6 * (1 - 0.75) * 0.80 + (1 - 0.6) * 0.75 * 0.5 )/(0.6 * (1 - 0.75) * 0.80 + (1 - 0.6) * 0.75 * 0.5 + 0.6 *0.75 * 0.98 + 0.4*0.25 * 0.20)

= 0.369357

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