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. If

then since

it follows that

If we use the formula

then we can substitute in the numbers

and solve

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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

.4/.6 = 2/3

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c) In order for them to be independent events we would have to have

P(T|H) = P(T)

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.

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