$\mathrm{P}, \mathrm{Q}$ and R try to hit the same target one after the other. If their probabilities of…
$\mathrm{P}, \mathrm{Q}$ and R try to hit the same target one after the other. If their probabilities of hitting the target are $\frac{2}{3}, \frac{3}{5}, \frac{5}{7}$ respectively, then the probability that the target is hit by P or Q but not by R is
$\frac{26}{105}$
$\frac{79}{105}$
$0$
$\frac{75}{105}$
Solution
Let $P=$ The event that $P$ hit target
$\mathrm{Q}=$ The event that Q hit target
$\mathrm{R}=$ The event that R hit target
$\because \mathrm{P}(\mathrm{P})=\frac{2}{3}, \mathrm{P}(\mathrm{Q})=\frac{3}{5}$ and $\mathrm{P}(\mathrm{R})=\frac{5}{7}$
Now, required probability
$\begin{aligned} & =\mathrm{P}(\mathrm{P}) \mathrm{P}\left(\mathrm{Q}^{\prime}\right) \mathrm{P}\left(\mathrm{R}^{\prime}\right)+\mathrm{P}\left(\mathrm{P}^{\prime}\right) \mathrm{P}(\mathrm{Q}) \mathrm{P}\left(\mathrm{R}^{\prime}\right)+\mathrm{P}(\mathrm{P}) \mathrm{P}(\mathrm{Q}) \mathrm{P}\left(\mathrm{R}^{\prime}\right) \\ & =\frac{2}{3} \times \frac{2}{5} \times \frac{2}{7}+\frac{1}{3} \times \frac{3}{5} \times \frac{2}{7}+\frac{2}{3} \times \frac{3}{5} \times \frac{2}{7}=\frac{8+6+12}{105}=\frac{26}{105}\end{aligned}$