2015 Cayley Problem 23

Nylah has her living room lights on a timer. Each evening, the timer switches the lights on randomly at exactly 7:00 p.m., 7:30 p.m., 8:00 p.m., 8:30 p.m., or 9:00 p.m. Later in the evening, the timer switches the lights off at any random time between 11 p.m. and 1 a.m. For example, the lights could be switched on at exactly 7:30 p.m. and off at any one of the infinite number of possible times between 11 p.m. and 1 a.m. On a given night, Nylah's lights are on for t hours. What is the probability that 4 < t < 5?

\textbf{(A)}\ \frac{1}{2}\quad \textbf{(B)}\ \frac{1}{4}\quad \textbf{(C)}\ \frac{2}{5}\quad \textbf{(D)}\ \frac{3}{10}\quad \textbf{(E)}\ \frac{7}{20}

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 Cayley contests here.