SOLUTION: Hi, I am mathematically challenged when it comes to probability on top of having a hard time with understanding English! A student received an "A" on the first test of the semes

Question 287817: Hi, I am mathematically challenged when it comes to probability on top of having a hard time with understanding English!
A student received an "A" on the first test of the semester. The student wants to calculate the probability of scoring an "A" on the second test. Historically,the instructor knows that the joint probability of scoring "A"s on the first two tests is 0.5. Also, historically, the probability that a student scores an "A" on the second test, given that a student scored an "A" on the first test. is 0.9. What is the probability that a student will score an "A" on the second test?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
The problem statement is:
A student received an "A" on the first test of the semester.
The student wants to calculate the probability of scoring an "A" on the second test.
Historically,the instructor knows that the joint probability of scoring "A"s on the first two tests is 0.5.
Also, historically, the probability that a student scores an "A" on the second test, given that a student scored an "A" on the first test, is 0.9.
What is the probability that a student will score an "A" on the second test?

There are two key statements in this problem.

The first key statement is:

Historically,the instructor knows that the joint probability of scoring "A"s on the first two tests is 0.5.

This means that, if we let x = the probability of scoring an A on the first test, and we let y = the probability of scoring an A on the second test, that:

p(x)*p(y) = .5

The second key statement is:

Also, historically, the probability that a student scores an "A" on the second test, given that a student scored an "A" on the first test, is 0.9.

This means that, once the student has scored an A on the first test, then the probability that the student will score an A on the second test is .9

This means that the equation:

p(x)*p(y) = .5 becomes:

p(x)*.9 = .5.

Divide both sides of this equation by .9 to get:

p(x) = (.5/.9)

This means that the probability that the student will score an A on the first test is (.5/.9).

Your joint probability becomes (.5/.9) * .9 = .5 which becomes .5 = .5 which is a true statement.

Since you are asked what the probability is that the student will score an A on the second test, then the answer has to be .9.

That does not have to be solved for. That is already given.

If you were asked what the probability is that the student will score an A on the first test, then the answer is (.9/.5) which is roughly equal to .555555555 which rounds to .56.

Now we'll look at confirmation for this answer.

Assume the answer is correct.

The probability that the student will score an A on the first test is (.5/.9).

The probability that the student will score an A on the second test, given that the student scored an A on the first test, is .9

The joint probability of the student scoring an A on the first test and the second test is .5

The joint probability is the probability that the student scored an A on the first test times the probability that the student scored an A on the second test.

That equals (.5/.9) * .9 = .5.

The joint probability is correct as given.

The probability that the student scores an A on the second test, given that the student scored an A on the first test, is .9 as given.

The probability that the student scores an A on the first test is (.5/.9) as calculated.

I believe there's a strong probability that the problem is worded wrong, and that the problem should be asking what is the probability that the student will get an A on the first test.

This is especially true if you were told that the answer should be .55 which is very close to .56.

The actual answer should be (.5/.9) and it should apply to the first test, not the second test.

The probability that the student gets an A on the second test has to be .9 because you are told:

1. That the student got an A on the first test.
2. That the probability that the student gets an A on the second test, given that the student got an A on the first test, is .9.

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