Question

Suppose you have to cross a train track on your commute. The probability that you will have to wait for a train is 1/5, or .20. If you don’t have to wait, the com mute takes 15 minutes, but if you have to wait, it takes 20 minutes.

What is the expected value of the time it takes you to commute?
Is the expected value ever the actual commute time? Explain.

Answer 1

So, the expected value of the time it takes to commute is 16 minutes.

Explanation:

Let X is time it takes you to commute. We calculate E(X)?

We know that the probability that you will have to wait for a train is 1/5, or .20. If you don’t have to wait, the com mute takes 15 minutes, but if you have to wait, it takes 20 minutes.

The probability that you will have to wait for a train is :

`p_2=0.2\\`

The probability of not waiting for a train is:

`p_1=1-p_2=1-0.2=0.8`

So we get the following probability distribution:

`P(X)=\left \{ {{0.8, \, \, \, 15} \atop {0.2, \, \, \, 20}} \right.`

We calculate E(X):

`E(X)=x_1\cdot p_1+x_2\cdot p_2\\\\E(X)=15\cdot 0.8+ 20\cdot 0.2\\\\E(X)=12+4\\\\E(X)=16`

So, the expected value of the time it takes to commute is 16 minutes.

Expected value is not always the actual travel time, because it can sometimes take anywhere from 15 to 20 minutes.