# statistics – 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.

1. 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.

2. 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?