Of the $700$ members of an organization, $120$ are lawyers. Two members will be selected at random. What is the probability that neither of the members selected will be a lawyer?
I know the answer is $(580\cdot 579)/(700\cdot 699)$ but I'm trying to use the complement rule to answer this question and I can't get the right answer. Why would the complement rule, $P(A)+P(\text{not }A)=1$, not work for this question?
Best Answer
The complement of "neither is a lawyer" is "at least one is a lawyer", so you need to calculate the probability of that. One way is to list the three cases: the first is a lawyer and the second is not; the first is not and the second is; they are both lawyers. These probabilities are disjoint, so can be added. You calculate each of them the same way you did the direct approach: for the first is a lawyer and the second is not, it is $\frac {120\cdot580}{700\cdot 699}$