Consider the midterm and final for a statistics class Suppose 10% students earned an A on the midterm. Of those students who earned an A on the midterm, 55% received an A on the final, and 15% of the students who earned lower than an A on the midterm received an A on the final. You randomly pick up a final exam and notice the student received an A. What is the probability that this student earned an A on the midterm?

Suppose 10% students earned an A on the midterm. Of those students who earned an A on the midterm, 55% received an A on the final, and 15% of the students who earned lower than an A on the midterm received an A on the final.

This means that

Of the 10% of students who earned an A on the midterm, 55% received an A on the final. Also, of the 90% who did not earn an A on the midterm, 15% received an A on the final.

So, the percentage of students who got an A on the final exam is

[tex]P_{A} = 0.10(0.55) + 0.90(0.15) = 0.19[/tex]

To find the probability that this student earned an A on the final test also earned on the midterm, we divide the percentage of students who got an A on both tests by the percentage of students who got an A on both tests.

The percentage of students who got an A on both tests is:

[tex]P_{AA} = 0.10(0.55) = 0.055[/tex]

The probability that the student also earned an A on the midterm is

[tex]P = \frac{P_{AA}}{P_{A}} = \frac{0.055}{0.19} = 0.29[/tex]

There is a 29% probability that this student earned an A on the midterm.