8. An instructor passed out midterm reports after the first test. The report contained a homework grade and a test grade. 30% of the class got unsatisfactory grades on both the homework and the test. 60% got satisfactory grades on the homework, and 50% got satisfactory grades on the test.
a) If the experiment is checking on student's grades, let H be the event that the student got a satisfactory grade on the homework, and let T be the event that the student got a satisfactory grade on the test. Let us make up a Venn diagram.
then since 30% of the homework and the tests, we put a .3 outside of the union. This tells us that
Pr(T ∪ H) = .7
When working with these Venn diagrams, it works best to start with the smallest set, T ∩ H and work out. Diabolically, T ∩ H is the one thing we do not know. So we use the formula
We can fill in numbers
.7 = .5 + .6 - Pr(T ∩ H)
and solve for Pr(T ∩ H).
Pr(T ∩ H) = .7 - .5 - .6 = .4
b) We can now fill in the Venn diagram.
The question is, what percent of the students who got satisfactory grades on the homework got satisfactory grades on the test? We can use the formula for conditional probability.
which we can now see comes out to be
c) In order for them to be independent events we would have to have
that is, the conditional probability would have to be the same as the original probability, but 2/3 is not equal to .5. In this case, the students' chances of doing well on the test improve if they do the homework.