There are two lines of defense against an attacking aircraft, the firs

Let's look at this from the perspective of the surviving aircrafts.

7 out of 9 aircrafts will survive the missile.
7 out of 8 aircraft will survive the gun.

That is 49 out of 72 aircrafts will survive

Aircrafts which would survive = 7/9 * 7/8 = 49/72

Aircrafts which will not survive = 1 - [49/72] = 23 / 72

That's 23 aircrafts out of every 72 aircrafts which won't survive.

So, answer is option A

Solution:

We use the complement rule to solve this problem. There are 4 possible outcomes in this scenario, where H = the aircraft is hit and N = the aircraft is not hit:

(H, H) means the aircraft is hit by both the missile and the gun
(H, N) means the aircraft is hit by the missile but not the gun
(N, H) means the aircraft is not hit by the missile but is hit by the gun
(N, N) means the aircraft is hit by neither the missile nor the gun

Note that any of the first 3 outcomes would indicate a hit. So only the fourth outcome (N, N) is the outcome where the aircraft is not hit. Thus, it is easier to calculate the probability of (N, N) and subtract this probability from 1 (using the rule of the complement) to find the probability that the aircraft is hit.

The probability of not hitting the aircraft by either the missile or the gun - (N, N) - is (1 - 2/9) x (1 - 1/8) = 7/9 x 7/8 = 49/72. Thus, the probability of hitting the aircraft is 1 - 49/72 = 23/72.

Answer: A

The listing of the Completely Exhaustive events:

P(Missile Hits and Gun Misses)

+

P(Missile Misses and Gun Hits)

+

P(BOTH Missile and Gun Hit)

+

P(NEITHER Hit)

=

1


1 - P(Neither Hit) = Probability that the plane is Hit



Or, the long way



P(Missile Hits and Gun Misses) = (2/9) (7/8) = 14/72

+

P(Missile misses and Gun Hits) = (7/9) (1/8) = 7/72

+

P(Missile Hits and Gun Hits) = (2/9) (1/8) = 2/72

14/72 + 7/72 + 2/72 = 23/72

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