What is the approximate probability that Cody chose to get home from work by bus, given that he arrived home after 7 p.m.?

Question

Suppose that Cody can choose to get home from work by car or bus.

 

1. When he chooses to get home by car, he arrives home after 7 p.m. 9 percent of the time.
2. When he chooses to get home by bus, he arrives home after 7 p.m. 18 percent of the time.
3. Because the bus is cheaper, he uses the bus 65 percent of the time.

 

What is the approximate probability that Cody chose to get home from work by bus, given that he arrived home after 7 p.m.?

12%

65%

18%

79%

Answer 1

You would use Bayes Theorem for this conditional probability:

Let B be the event that he takes the bus, C the event that he takes a car, and L the event that he is late.

We are looking for the probability that he takes the bus given that he is late. In probability notation it looks like:

P( B | L )

Bayes says: P( B | L ) = P (B ∩ L) / P (L)

I recommend you try work this out on your own without looking forward first.

Here is the working out:

P (B ∩ L) = .65 * .18 (intuitively it is the probability that he takes the bus and is late)

P(L) = P( B ∩ L) + P(C ∩ L) = .65 * .18 + .35 * .09

(intuitively, the probability that he takes the bus and is late, or takes a car and is late)