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Problem 183

Combinatorics Difficulty 1.5 Multiple choice CEMC Pascal · Canada · 2023

Each of four doors is randomly either open or closed. What is the
probability that exactly two of the four doors are open?

Pick one

Official solution

There are 2 possible “states” for each door: open or
closed.

Therefore, there are $2 $\times 2 \times 2 \times$ 2 = 2^4 = 16$ possible combinations of open and closed
for the 4 doors.

If exactly 2 of the 4 doors are open, these doors could be the 1st and
2nd, or 1st and 3rd, or 1st and 4th, or 2nd and 3rd, or 2nd and 4th, or
3rd and 4th. Thus, there are 6 ways in which 2 of the 4 doors can be
open.

Since each door is randomly open or closed, then the probability that
exactly 2 doors are open is $\frac{6}{16}$ which is equivalent to $\frac{3}{8}$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.