Question

On your way back from a family vacation, my mom refuses to take the expressway home. She says, “There could be bumper to bumper traffic on the expressway, and even if there isn't, there's a good chance there will be some traffic and it will take the same time as taking the side roads anyway.”

There are only three options for traffic: heavy, moderate, and no traffic. These are mutually exclusive and jointly exhaustive.

Suppose that, with no traffic, the expressway will get you home in 2 hours. But there is a 20% chance that you will encounter heavy traffic on the expressway, in which case it will take 6 hours.

And there is a 40% chance you will encounter moderate traffic, in which case it will take 4 hours.

Taking side roads will get you home in 4 hours with no traffic at all. But there is a 20% chance of moderate traffic on these roads, which will make for a 5 hour trip, and a 10% chance of heavy traffic, which will make for a 6 hour trip.

What is the difference in expected time between taking the expressway and taking the side roads?

Give an answer as a positive value in hours.

(Note: expected *time* is like expected monetary value, except that you multiply the probability of each outcome by the time spent, rather than by the monetary value. Just do everything in terms of fractions of hours, and don't convert to minutes!)

Answer 1