2012 Pascal Problem 25

There are four people in a room.

For every two people, there is a 50% chance that they are friends.

Two people are *connected* if:

- they are friends, or - a third person is friends with both of them, or - they have different friends who are friends of each other.

What is the probability that every pair of people in this room is connected?

\textbf{(A)}\ \frac{18}{32}\quad \textbf{(B)}\ \frac{20}{32}\quad \textbf{(C)}\ \frac{22}{32}\quad \textbf{(D)}\ \frac{19}{32}\quad \textbf{(E)}\ \frac{21}{32}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.