Question

A subway has good service 70% of the time and runs less frequently 30% of the time because of signal problems. When there are signal problems, the amount of time in minutes that you have to wait at the platform is described by the pdf probability density function with signal problems = pT|SP(t) = .1e −.1t But when there is good service, the amount of time you have to wait at the platform is probability density function with good service = pT|Good(t) = .3e −.3t You arrive at the subway platform and you do not know if the train has signal problems or is running with good service, so there is a 30% chance the train is having signal problems. (a) What is the probability that you wait at least 1 minute if there is good service? (b) What is the probability that you wait at least 1 minute if there are signal problems? (c) After 1 minute of waiting on the platform, you decide to re-calculate the probability that there are signal problems conditioning on the fact that your wait will be at least 1 minute long (since you have already waited 1 minute). What is that new probability? (d) After 5 minutes of waiting, still no train. You re-calculate again. What is the new probability?

Answer 1

Let

• SP= event that the train is having signal problems
• Good=event that the train has good service

There is a 30% chance the train is having signal problems is same as the probability that the train has signal problems is 0.30

P(SP) = 0.30

A subway has good service 70% of the time is same as the probability that the train has good service is 0.70

P(Good)=0.70

Let T be the amount of time in minutes that you have to wait at the platform.

The conditional pdf of T when there is a signal problem is

The conditional pdf of T when there is good service is

(a) The probability that you wait at least 1 minute if there is good service is same as the conditional probability that you wait at least 1 minute given that there is good service

ans: the probability that you wait at least 1 minute if there is good service is 0.7408

(b) The probability that you wait at least 1 minute if there are signal problems is same as the conditional probability that you wait at least 1 minute given that there are signal problems

ans: The probability that you wait at least 1 minute if there are signal problems is 0.9048

(c) After 1 minute of waiting on the platform, you decide to re-calculate the probability that there are signal problems conditioning on the fact that your wait will be at least 1 minute long (since you have already waited 1 minute).

The new probability that there are signal problems given that our wait will be at least 1 minute long is

ans: the probability that there are signal problems conditioning on the fact that your wait will be at least 1 minute long is 0.3436

(d) After 5 minutes of waiting, still no train. You re-calculate again.

the conditional probability that you wait at least 5 minute given that there is good service

the conditional probability that you wait at least 5 minute given that there are signal problems

The new probability that there are signal problems given that our wait will be at least 5 minute long is

ans: the probability that there are signal problems conditioning on the fact that your wait will be at least 5 minute long is 0.5381