Probability

To find the probability that the target is hit by $P$ or $Q$ but not by $R$, let's define the events:$E_1: P$ hits the target.$E_2: Q$ hits the target.$E_3: R$ hits the target.The given probabilities are:$\;P(E_1) = \;\frac{2}{3}$$\;P(E_2) = \;\frac{3}{5}$$\;P(E_3) = \;\frac{5}{7}$These events are independent, so:The probability that $P$ hits and both $Q$ and $R$ miss is:$P(E_1 \cap \overline{E_2} \cap \overline{E_3}) = P(E_1) \cdot (1-P(E_2)) \cdot (1-P(E_3)) = \;\frac{2}{3} \cdot \;\frac{2}{5} \cdot \;\frac{2}{7}$The probability that $Q$ hits and both $P$ and $R$ miss is:$P(\overline{E_1} \cap E_2 \cap \overline{E_3}) = (1-P(E_1)) \cdot P(E_2) \cdot (1-P(E_3)) = \;\frac{1}{3} \cdot \;\frac{3}{5} \cdot \;\frac{2}{7}$The probability that both $P$ and $Q$ hit but $R$ misses is:$P(E_1 \cap E_2 \cap \overline{E_3}) = P(E_1) \cdot P(E_2) \cdot (1-P(E_3)) = \;\frac{2}{3} \cdot \;\frac{3}{5} \cdot \;\frac{2}{7}$Now, summing these probabilities gives the required probability that the target is hit by $P$ or $Q$, but not by $R$ :$P(\;\text{ hit by }\;P\;\text{ or }\;Q\;\text{ but not }\;R)\;=P(E_1 \cap \overline{E_2} \cap \overline{E_3}) + P(\overline{E_1} \cap E_2 \cap \overline{E_3}) + P(E_1 \cap E_2 \cap \overline{E_3})$$\;=\;\frac{2}{3} \cdot \;\frac{2}{5} \cdot \;\frac{2}{7}+\;\frac{1}{3} \cdot \;\frac{3}{5} \cdot \;\frac{2}{7}+\;\frac{2}{3} \cdot \;\frac{3}{5} \cdot \;\frac{2}{7}$$\;=\;\frac{8}{105}+\;\frac{6}{105}+\;\frac{12}{105}$$\;=\;\frac{26}{105}$Thus, the probability that the target is hit by $P$ or $Q$ but not by $R$ is $\;\frac{26}{105}$.