Question

Assume that the probability that a bomb dropped from an aeroplane will strike a certain target is 0.2. If 6 bombs are dropped, find the probability that
(i) exactly 2 will strike the target
(ii) at least 2 will strike the target

Answer 1

Let X be the number of bombs that hit the target.
Then, X follows a binomial distribution with n = 6
Let p be the probability that a bomb dropped from an aeroplane will strike the target.

$$\begin{gathered} \therefore \mathit{}p=0.2\mathrm{and}q=0.8 \\ \mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{6}C_r{\left(0.2\right)}^r{\left(\mathit{0}\mathit{.}\mathit{8}\right)}^{\mathit{6}\mathit{-}r} \\ \left(\mathrm{i}\right)P(\mathrm{exactly}2\mathrm{will}\mathrm{strike}\mathrm{the}\mathrm{target})=P(X=2) \\ ={}^{6}C_2(0.2{)}^2(0.8{)}^4 \\ =15(0.04)\left(0.4096\right) \\ =0.2458 \\ \left(\mathrm{ii}\right)\mathit{}P(\mathrm{at}\mathrm{least}2\text{will}\mathrm{strike}\mathrm{the}\mathrm{target})=P(X\ge 2) \\ =1-\left[P\right(X=0)+P(X=1\left)\right] \\ =1-(0.8{)}^6-6(0.2\left)\right(0.8{)}^5 \\ =1-0.2621-0.3932 \\ =0.3447 \end{gathered}$$