Let us look at some examples below that go through calculating the probabilities of being between $x$-values.

Example 34.1.1

Problem Setup: During peak periods, the times between 130 busses arriving at BCIT follow a continuous uniform distribution with a minimum of 2 minutes and a maximum of 12 minutes.

Question: What is the probability that it takes between 5 to 10 minutes for the next bus to arrive?

Solution: This gives the following values:

• $a=2$
• $b=12$
• $x_1 = 5$
• $x_2 = 10$

Plugging this all into the formula gives: $$P(5\le x \le 10) = \frac{x_2-x_1}{b-a} = \frac{10-5}{12-2} = 0.5$$

Conclusion: There is a 50% chance that the 130 bus will take between 5 to 10 minutes to arrive.

Let us now change up the problem slightly and show the solution in a video (see below).

Example 34.1.2

Problem Setup: Your friend was waiting for you at the bus stop and texted that you JUST missed the bus. You are hoping to join them to watch a movie. If you can catch a bus in the next 10 minutes, you will make it the movie on time.

• It takes you 2 minutes to pack up your stuff.
• It takes 1 minute to run to the bus stop.
• The times between busses are between 2 and 12 minutes
• The times follow a continuous uniform distribution

Question: What is the probability of making it to the movie on time?

Solution: Click here to download the written solutions. Also, see the video below:

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