How to determine whether to use complement of event when calculating probability? A sample problem in my textbook states: In a recent year, there were 18,187 U.S. allopathic medical school seniors who applied to residency programs and submitted their residency program choices. Of these seniors, 17,057 were matched with residency positions, with about 79.2% getting one of their top three choices. Medical students rank the residency programs in their order of preference, and program directors in the United States rank the students. The term “match” refers to the process whereby a student’s preference list and a program director’s preference list overlap, resulting in the placement of the student in a residency position.

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*Find the probability that a randomly selected senior was matched with a residency position and it was one of the senior’s top three choices.


*Find the probability that a randomly selected senior who was matched with a residency position did not get matched with one of the senior’s top three choices.
1 is obvious: $(\frac{17057}{18187})(0.792) \approx 0.743$
For 2, the textbook says to take the complement of 0.792: $1 - 0.792 = 0.208$
But my question is: why wouldn't you solve this one the same way as part 1? As in:
$$(\frac{17057}{18187})(1 - 0.792) \approx 0.195$$
why wouldn't you do that?
 A: The second question says "a randomly selected senior who was matched with a residency position".
So the randomly selected senior is one of the "17057". Not anyone of the "18187". That's why the answer is simply $1-0.792 = 0.208$. Because for these 17057 seniors (who were matched with a residency position) we know that the $79.2\%$ of them got one of their top three choices.
(You can see that if in your way of calculating the probability change the denominator to 17057 (the size of the population from where we sample the student), the result is $0.208$)
Generally speaking, and answering more to the title of your question rather than the specific exercise, you just use complement of event when it is easier (easier usually means fewer calculations) or when it is the obvious way of solving an exercise (like here).
A: Basically what they have done is-
Let us define an event A and B-A: Matching with one of the senior’s top three choices.
$$P(A)=\frac{79.2}{100}$$
B: Senior was matched with a residency position.
$$P(B)=\frac{17057}{18187}$$
1st part says you to find-
$$P(B \cap A)=P(B).P(A)=\frac{17057}{18187}.\frac{79.2}{100}$$
2nd part says you to find $$P(A'|B)=\frac{P(A' \cap B)}{P(B)}=\frac{P(A').P(B)}{P(B)}=P(A')=1-0.792=0.208$$
The important thing here is the keyword 'who' used in second statement, that results this in a conditional probability. 
Note- $P(A' \cap B)=P(A').P(B)$ as these are independents events.
