Question

The probability that A hits a target is 1/3 and the probability that B hits it, is 2/5, What is the probability that the target will be hit, if each one of A and B shoots at the target?

Answer 1

$$\begin{gathered} P\left(A\right)=P\left(\text{A hits target}\right)=\frac{1}{3} \\ P\left(B\right)=P\left(\text{B hits target}\right)=\frac{2}{5} \\ \text{Now}, \\ P\left(A\cup B\right)=P\left(\text{target will be hit by either A or B}\right) \\ \Rightarrow P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ \Rightarrow P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\right)P\left(B\right)\left[\text{A and B are independent}\right] \\ \Rightarrow P\left(A\cup B\right)=\frac{1}{3}+\frac{2}{5}-\frac{1}{3}\times \frac{2}{5} \\ \Rightarrow P\left(A\cup B\right)=\frac{5+6-2}{15} \\ \Rightarrow P\left(A\cup B\right)=\frac{9}{15} \\ \Rightarrow P\left(A\cup B\right)=\frac{3}{5} \end{gathered}$$