Assume that 60% of the students at Remmington

Answered question

2022-04-01

Assume that 60% of the students at Remmington High studied for their Psychology test.  Of those that studied, 25% got an A, but only 8% of those who didn't study got an A.  What is the approximate probability that someone that gets an A actually studied for the Psychology test?

 

Answer & Explanation

alenahelenash

alenahelenash

Expert2023-04-27Added 556 answers

Let's define the following events:
- S: the event that a student studied for their Psychology test.
- A: the event that a student got an A in their Psychology test.
We are given that P(S)=0.6 (i.e., 60% of the students studied), P(A|S)=0.25 (i.e., among those who studied, 25% got an A), and P(A|¬S)=0.08 (i.e., among those who did not study, 8% got an A). We want to find P(S|A), the probability that a student studied given that they got an A.
We can use Bayes' theorem to find P(S|A):
P(S|A)=P(A|S)P(S)P(A)
We can find P(A) using the law of total probability:
P(A)=P(A|S)P(S)+P(A|¬S)P(¬S)
We can compute P(¬S) as 1P(S):
P(¬S)=1P(S)
Substituting these values, we get:
P(A)=0.25×0.6+0.08×0.4=0.17
Now we can substitute P(A) and P(A|S) into Bayes' theorem to get:
P(S|A)=0.25×0.60.170.882
Therefore, the approximate probability that someone who got an A actually studied for the Psychology test is 0.882, or about 88.2%.

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