Question

Your friend is studying for an exam. Based on your knowledge of your friend, you believe that if they study for the exam, there is an 80% probability they will be able to pass it. On the other hand, if they do not study, there is only a 30% probability they will be able to pass. Your friend is not a particularly industrious student, and you initially believe there is only a 60% probability your friend will study for the exam. A few days later your friend happily proclaims that they passed the exam. Thus, find the probability that they did in fact study for the test with this knowledge in hand.

Answer 1

0.8 = 80% probability that they did in fact study for the test.

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Passed the exam.

Event B: Studied.

Probability of passing the test:

80% of 60%(Studied).

30% of 100 - 60 = 40%(did not study). So

`P(A) = 0.8*0.6 + 0.3*0.4 = 0.6`

Probability of passing the test studying:

80% of 60%, so:

`P(A \cap B) = 0.8*0.6 = 0.48`

Find the probability that they did in fact study for the test with this knowledge in hand.

`P(B|A) = (P(A \cap B))/(P(A)) = (0.48)/(0.6) = 0.8`

0.8 = 80% probability that they did in fact study for the test.