Question

Bob is running late for his class and missed his bus. He calls his friend Sam to ask for a ride to school.

Sam, tired of Bob interrupting his sleep every week, has lowered the volume of his phone. There is a 70% chance Sam sleeps through Bob's phone call and does not answer this morning.

If Sam answers the phone, he will drive to Bob's house and give Bob a ride. Bob will get to school in X minutes, where X follows a (continuous) Uniform(15,25) distribution.

If Sam does not answer the phone, Bob has to walk to school. Bob is a consistent walker, and will always make it to school in exactly 40 minutes.

What is the expected time for Bob to get to school today (in minutes)?

Answer 1

70% chance Sam sleeps through Bob's phone call and does not answer this morning

Probability of Sam not answering the Bob's call = 0.70

Probability of Sam answering the call = 1-0.70=0.30

Scenario 1:

If Sam does not answer Bob's call this morning then makes it to school exactly 40 minutes .

Probability of Sam does not answer Bob's call this morning = 0.70

Probability of scenario 1 = 0.7

For this Scenario 1, Expected value for Bob to get to school today (in minutes) = 40

Scenario 2:

If Sam answers Bob's call this morning then makes it to school X minutes;   X follows a (continuous) Uniform(15,25) distribution

Probability of Sam does not answer Bob's call this morning = 0.30

Probability of scenario 2 = 0.30

For this scenario 2, Expected time for Bob to get to school today is the Expected Value of X

Expected Value of Uniform distribution (a,b) = (a+b)/2 i.e E(X) = (15+25)/2=40/2=20

For this scenario 2, Expected value for Bob to get to school today (in minutes) = 20

Probability of Scenario 1 = 0.7 and for this scenario the Expected time for Bob to get to school today (in minutes) =40

Probability of Scenario 1 = 0.3 and for this scenario the Expected time for Bob to get to school today (in minutes) =20

Expected time for Bob to get to school today (in minutes) = 0.7 x 40 + 0.3 x 20 = 28+6 = 34

Expected time for Bob to get to school today (in minutes) = 34