2 Late for Clas One of the student attending RMI 3750 leaves from North Springs

Question

2 Late for Clas One of the student attending RMI 3750 leaves from North Springs everyday to come to class. He can choose to commute using his car (C), the bus (B) or the MARTA train (T). Due to heavy traffic, any day he uses his car to come to school there is 80% probability that he will be late, when he uses the bus, the probability of being late is 40% and when he rides the train the probability of being late is only 3%. (1) Suppose that, as the instructor, I do not know how this student commutes to class (that is, I assume the probability that he uses any means of transportation is equally likely). One day, I notice that the student is late. What is my estimate of the probability that. a.... the student used his car that day? the student used the bus that day? the student took the train that day? b. c. (2) Suppose that this student is your friend and you know that he is fourth more likely to use his car than the bus. Furthermore, you know that the probability that he rides the train is 25%, when the student is late, what is your estimate of the probability that the student took the train that day?

Explanation / Answer

1)

a)X = the student is late
P(C|X) = P(C and X)/P(X)
P(X) = P(X|C)P(C) + P(X|B)P(B) + P(X|T)P(T)
= 1/3 ( 0.8 + 0.4 + 0.03)
= 1/3 * 1.23

hence
P(C|X) = P(C and X)/P(X)
= P(X|C)P(C) / P(X)
= 1/3 * 0.8 / (1/3 * 1.23)
= 0.8/1.23

similarly
b) P(B|X) = 0.4/1.23
c) P(T|X) = 0.03 /1.23

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