Probability of Leaving Early Given Lateness: A Bayes' Theorem Problem

We can use Bayes' theorem to solve this problem:

Let A be the event that Pablo leaves home before 07:00, and B be the event that he is late for work. We want to find P(A|B), the probability that Pablo left home before 07:00 given that he is late for work.

Using the law of total probability, we can calculate the probability of being late for work as:

P(B) = P(A)P(B|A) + P(A')P(B|A')

where A' is the complement of A (i.e. the event that Pablo leaves home at 07:00 or later).

Substituting the given probabilities, we have:

P(B) = (3/4)(1/8) + (1/4)(5/8) = 11/32

Now we can use Bayes' theorem to find P(A|B):

P(A|B) = P(A)P(B|A) / P(B)

Substituting the given probabilities, we have:

P(A|B) = (3/4)(1/8) / (11/32) ≈ 0.068

Therefore, the probability that Pablo left home before 07:00 given that he is late for work is approximately 0.068, or about 6.8%.