At a college, 72% of courses have final exams and 46% of courses require research papers. suppose that 32% of courses have a research paper and a final exam. let f be the event that a course has a final exam. let r be the event that a course requires a research paper. a. find the probability that a course has a final exam or a research project. b. find the probability that a course has neither of these two requirements.

a)

b)

Step-by-step explanation:

Let's define the following events first:

F: The event that a course has a final exam.

R: The event that a course requires a research paper

From the info provided we have that:

P(F and R) =0.32

So then we can create a Venn diagram as we can see on the figure attached.

a. Find the probability that a course has a final exam or a research project.

For this case we can find the probability like this:

b. Find the probability that a course has NEITHER of these two requirements.

For this case we can use the complement rule and we can find the probability like this:

And that's the same value obtained with the diagram.

a precentage of 6th graders. the population is all 6th graders. let me know if i'm wrong, i hope this ! : )

step-by-step explanation: