Cumulative Probability: prob. rider waits >= 50 min for bus

I am really confused on how to do the cumulative probability. I have a few homework problems dealing with it and I am totally stuck. One question is:

Suppose that the time students wait for a bus can be described by a uniform random variable X, where X is between 15 minutes and 85 minutes.

(a) What is the probability P that a student will wait between 15 and 50 minutes for the next bus?

(b) What is the probability P that a student will have to wait at least 50 minutes for the next bus?

Re: Cumulative Probability

If X is a uniform random variable, the probability for all events in that range (15 to 85 minutes) is the same. The density function looks like a rectangle. The total probability over the whole range is 1. Let's say we're talking about a discrete uniform distribution, where five events have the same probability of occurring:

We put five cards in a hat, a 2, 3, 4, 5, and 6. What is the likelihood we will pick out at least a 5?

Each of those events has an equal probability, which is 1/5. The cumulative probability is simply the sum of every event that will meet the problem's conditions, in this case, 2/5.

Now, with yours, I'm not sure how many events we're talking about (because you didn't say they arrive in 5, 10, or 15-minute intervals). Maybe they arrive every minute, in which case, we're talking about 71 events, each of which has exactly the same probability of occurring. That is what determines the width of your rectangle, and the height is such that width times height = 1. In order to solve the problem, you need to know how many positive events there are and how many total events there are. That will give you the fraction of your rectangle that shows the positive events, say how many of those events are greater than 50 minutes? That is, what is the area of the rectangle from 50 <= x <= 85?

If the distribution of X is continuous, you could use the cumulative distribution and the a and b parameters for the Beta (\(\displaystyle \beta\)) distribution:

\(\displaystyle f_X(x) = \frac{1}{b - a}, \quad \quad x \in [a, b]\)

Thank you for your help.