Q.Suppose the chances of hitting a target by a person $X$ are 3 times in 4 shots, by $Y$ are 4 times in 5 shots, and by $Z$ are 2 times in 3 shots. They fire simultaneously exactly one time each. What is the probability that the target is damaged by exactly 2 hits?

Definition 12.15 (independence). Two events $A,B$ are independent exactly when

When $P(A),P(B)>0$, this is exactly equivalent to $P(B/A)=P(B)$ and $P(A/B)=P(A)$ -- knowing $A$ happened tells you nothing new about $B$'s chances, and vice versa. This is the special case of the Multiplication Theorem where the conditional term collapses to the plain probability. Mutually independent events $A_1,\ldots,A_n$ satisfy the stronger requirement that EVERY sub-collection multiplies out: $P(A_{i_1}\cap\cdots\cap A_{i_k})=P(A_{i_1})\cdots P(A_{i_k})$.

Theorem 12.8. If $A,B$ are independent, then so are (i) $\bar A,B$; (ii) $A,\bar B$; (iii) $\bar A,\bar B$. (Proved via De Morgan's law: $P(\bar A\cap\bar B)=P(\overline{A\cup B})=1-P(A\cup B)=1-[P(A)+P(B)-P(A)P(B)]=[1-P(A)][1-P(B)]=P(\bar A)P(\bar B)$.) …