For an airplane to be destroyed by a shell exploding at some…

For an airplane to be destroyed by a shell exploding at some distance from it, it is necessary that both engines are destroyed by shell fragments or that the pilot is killed. If a shell explodes at distance $r$ from the airplane, the probability that both engines are destroyed is $0.2$, and the probability that the pilot is killed is $0.3$. Find the probability that the airplane is knocked out by the explosion.

Step 1: Events

$E$ = "both engines destroyed", $P(E)=0.2$; $F$ = "the pilot is killed", $P(F)=0.3$; assumed independent.

Step 2: Addition rule

\[ \boxed{P(E+F)=0.2+0.3-0.2\cdot0.3=0.44.} \]

Step 3: Check with the opposite event

The airplane survives only if the engines are not both destroyed and the pilot survives: $0.8\cdot0.7=0.56$. So $P=1-0.56=0.44$ ✓.

Step 4: Remark

Simply adding $0.2+0.3=0.5$ would count twice the case in which both happen. The answer relies on independence; if, say, the two events were mutually exclusive, the answer would be $0.5$.