Four persons can hit a target correctly with probabilities \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\) and \(\frac{1}{5}\) respectively. If all hit at the target independently, then the probability that the target would be hit, is

Let event A : A can hit the target event B : B can hit the target event \(C\) : \(C\) can hit the target event D : D can hit the target
\(\mathrm{P}(\mathrm{~A})=\frac{1}{2}, \mathrm{P}(\mathrm{~B})=\frac{1}{3}, \mathrm{P}(\mathrm{C})=\frac{1}{4}, \mathrm{P}(\mathrm{D})=\frac{1}{5}\)
P (Target is not hit by any one of them)
\(=\mathrm{P}\left(\mathrm{~A}^{\prime} \cap \mathrm{B}^{\prime} \cap \mathrm{C}^{\prime} \cap \mathrm{D}^{\prime}\right)\)
\(=\mathrm{P}\left(\mathrm{A}^{\prime}\right) \cdot \mathrm{P}\left(\mathrm{B}^{\prime}\right) \cdot \mathrm{P}^{\prime}\left(\mathrm{C}^{\prime}\right) \cdot \mathrm{P}\left(\mathrm{D}^{\prime}\right)\)
\(=\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)\)
\(=\frac{1}{5}\)
\(\begin{aligned} & \mathrm{P}(\text { Target is hit }) \\ & =1-\mathrm{P}(\text { Target is not hit by any one of them }) \\ & =1-\frac{1}{5} \\ & =\frac{4}{5}\end{aligned}\)