Bill commutes to work in the business district of Boston. He takes the train 3/5 of the time and drives 2/5 of the time (when he visits clients). If he takes the train, then he gets home by 6:30 p. m. 83% of the time; if he drives, then he gets home by 6:30 p. m. 62% of the time. If Bill gets home by 6:30 p. m., what is the probability that he drove to work? (Round your answer to three decimal places.)

The probability that Bill drove to work given that he reached home by 6:30 PM is 0.3324.

Step-by-step explanation:

Denote the events as follows:

T = Bill takes the train to work

D = Bill drives to work

X = Bill gets home from work by 6:30 PM.

The information provided is:

P (T) = 0.60

P (D) = 0.40

P (X|T) = 0.83

P (X|D) = 0.62

The Bayes's theorem states that the conditional probability of an event Ei given that another event has already occurred is:

Use the Bayes' theorem to determine the value of P (D|X) as follows:

Thus, the probability that Bill drove to work given that he reached home by 6:30 PM is 0.3324.

step-by-step explanation: