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John drives and there are traffic lights on his routes. When all traffic lights on his...

John drives and there are traffic lights on his routes. When all traffic lights on his route are green, the entire trip takes 18 minutes. JohnŠs route includes 5 traffic lights, each of which is red with probability 1/3, independent of every other light. Each red traffic light that he encounters adds 1 minute to his commute. (a) Find the expectation, and variance of the length (in minutes) of John’s commute. (b) What is the pmf of the length (in minute) of John’s commute? (c) Given that John’s commute took him at most 19 minutes, what is the expected number of red lights that he encountered? (d) Given that John encountered a total of three red lights, what is the probability that exactly two out of the first three lights were red?

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