Problem:
A person drives to work via a road with a single traffic signal. The light cycles, green for 45 seconds, red for 15 seconds – ignore yellow. Assume the person approaches the signal at a random time. Please find:
a. the probability of having to stop at the light;
b. the expected delay time due to the signal.
Attempt at Solution:
I was able to deduce that given the total 60 seconds, the probability of the light being green is 75% and the probability of the light being red is 25%. This, however, is where I am stuck.
So I imagine you concluded that the probability of having to stop is $\frac{1}{4}$.
Now to the expected delay. If the light is green, the expected delay is $0$. If the light is red, by symmetry the average delay is $7.5$ seconds. Thus the average delay, in seconds, is $$(3/4)(0)+(1/4)(7.5).$$