12.05 A study of the students taking distance learning courses at a university finds that they are mostly older students not living in the university town. Choose a distance learning student at random. Let A be the event that the student is 25 years old or older and B the event that the student is local. The study finds that:

P(A) = 0.7, P(B) = 0.25, and P(A and B) = 0.05. What is the probability that the student is at least 25 years old and not local?

To find the probability that the student is at least 25 years old and not local, we need to find the probability of event A occurring and event B not occurring.

Let's start by finding the probability of event B not occurring, which is the complement of event B. We can denote event B' as the event that the student is not local.

P(B') = 1 - P(B) = 1 - 0.25 = 0.75

Next, we need to find the probability of event A occurring and event B' occurring. We can denote event A and B' as the event that the student is at least 25 years old and not local.

P(A and B') = P(A) - P(A and B) = 0.7 - 0.05 = 0.65

Therefore, the probability that the student is at least 25 years old and not local is 0.65.

To find the probability that the student is at least 25 years old and not local, we need to find the probability of event A (the student is 25 years old or older) and event B (the student is local) not both happening at the same time. In other words, we want to find P(A and not B).

The formula for finding the probability of two events not both happening is: P(A and not B) = P(A) - P(A and B).

Given that P(A) = 0.7, P(B) = 0.25, and P(A and B) = 0.05, we can substitute these values into the formula:

P(A and not B) = P(A) - P(A and B) = 0.7 - 0.05 = 0.65

Therefore, the probability that the student is at least 25 years old and not local is 0.65.