Question

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are $ 0.3 $ and $ 0.2 $ respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is
A. $ 0.06 $
B. $ \dfrac{7}{{22}} $
C. $ 0.2 $
D. $ 0.7 $

Answer 1

Hint: Probability is the state of being probable and the extent to which something is likely to happen in the particular situations or the favourable outcomes. Probability of any given event is equal to the ratio of the favourable outcomes with the total number of the outcomes. Here we will consider the probability of aeroplane II hitting the target in first turn, second turn and so on. We will use the concept of the geometric progression.

Complete step-by-step answer:
Let us assume that the probability of aeroplane I hitting correctly be $ P(A) $
 $ \therefore P(A) = 0.3 $ $ $ (Given)
Also, us assume that the probability of aeroplane II hitting correctly be $ P(B) $
 $ \therefore P(B) = 0.2 $ $ $ (Given)
The probability of aeroplane II hitting the target in the first turn can be given by $ P(\overline A ) \times P(B) $
Hence, the probability of aeroplane II hitting the target is –
 $ P(\overline A ) \times P(B) + P(\overline A ) \times P(\overline B ) \times P(\overline A ) \times P(B) + .... $
Place the values in the above equation –
 $ \Rightarrow 0.7 \times 0.2 + 0.7 \times 0.8 \times 0.7 \times 0.2 + ..... $
Simplify the above equation –
 $ \Rightarrow 0.14 + 0.14(0.56) + 2(0.14)(0.56) + .... $
Take out the common multiple from the above equation –
 $ \Rightarrow 0.14[1 + 0.56 + 2(0.56) + ....] $
The above equation is the geometric progression- $ r = 0.56,{\text{ a = 1}} $
So, the required probability becomes –
 $ = 0.14\left[ {\dfrac{1}{{1 - 0.56}}} \right] $
Simplify the above equation –
 $
   = 0.14\left[ {\dfrac{1}{{0.44}}} \right] \\
   = \dfrac{{14}}{{44}} \;
  $
Take out common multiples from the numerator and the denominator –
 $ = \dfrac{7}{{22}} $
So, the correct answer is “Option B”.

Note: Be good in multiples and the simplification of the fractions. The probability of any event always ranges between zero and one. It can never be the negative number or the number greater than one. The probability of impossible events is always equal to zero whereas, the probability of the sure event is always equal to one.