What is the probability of two people who hit with probabilities p and q respectively neither hitting their targets?
A is the event that person one hits ~ Ber(p) B is the event that person two hits ~ Ber(q).
I thought the answer is $1-p \cdot q = P((A \cap B)^c)=p((A^C \cup B^c) = P(A^C)+P(B^C) - p(A^C \cap B^C) = (1-p)+(1-q) - (1-p)*(1-q)$
The textbook says $(1-p)(1-q)$
Without giving it away (because this should be tagged as self-study, the problem was in this step:
$$1-p \cdot q = P((A \cap B)^c)\text{.}$$
On the right, you’ve written “the probability of not (both A and B),” which isn’t what you intended. That’s either one of them—or both—missing, not both of them missing—which is what you want. The latter is equivalent to “neither of them hitting”; the former is equivalent to “at most one of them hitting”.
