Every morning Mary randomly decides on one of three possibleways to get to work. She makes her choice so that all three choicesare equally likely. The three choices are described as follows: •Choice A (Drives the highway): The highway has no traffic lightsbut has the possibility of accidents. The number of accidents onthe highway for the hour preceding Mary’s trip, X, follows aPoisson distribution with an average of 2. The time (minutes) ittakes her to get to work is affected by the number of accidents inthe hour preceding her trip due to clean up. The time (in minutes)it takes her is given by T = 54.5 + 5X. • Choice B (Drives throughtown): Suppose there is no possibility of being slowed down byaccidents while going through town. However, going through town shemust pass through 10 traffic lights. Suppose all traffic lights actindependently from one another and for each there is a probabilityof 0.5 that she will have to stop and wait (because it is red). LetY be the number of lights she will stop and wait at. The time (inminutes) it takes her is given by T = 58.5 + Y. • Choice C (Takesthe train): Trains arrive for pick-up every 5 minutes. If the trainhas room, it will take her exactly 50 minutes to get to work. If anarriving train is full she will have to wait an additional 5minutes until the next train arrives. Trains going through thestation will arrive full with probability 0.75, and thus she cannotget on and will have to wait until the next train. Suppose it takesMary exactly 5 minutes to get to the train station and she alwaysarrives at the station just as a train arrives. Let Z be the numberof trains she’ll see until she can finally board (the train isn’tfull). The time (in minutes) it takes her is given by T = 50 +5Z.
a) Which choice should she make every morning tominimize her expected travel time?
b) On one morning Mary starts her journey to work at7am. Suppose it is necessary that she is at work at or before 8:00am. Which route should she take to maximize the probability thatshe is at work at or before 8:00am?
