Problem:
As a good test taker, on a multiple choice exam with 5 options per question, Gabe either knows the answer to a question beforehand or chooses an answer completely at random.
If he knows the answer beforehand, he selects the correct answer.
The probability that Gabe knows the answer to any given question is 0.6. Find the probability that an answer that was correct was one for which he knew the answer.
There are two possible events, Gabe either knows the answer or he doesn’t.
table = pd.DataFrame(index=["Knows", "Doesn't know"])
Using the information provided in the problem, we can fill in the table as follows:
table["prior"] = 0.6, 0.4
Now the likelihood is the probability of getting the correct answer given that Gabe knows the answer or doesn’t know the answer.
table["likelihood"] = 1, 1/5
Then we can work out the posterior probability of Gabe knowing the answer given that he got the answer correct.
table["posterior"] = table["unnorm"] / table["unnorm"].sum()
The probability of Gabe knowing the answer given that he got the answer correct is $\approx 0.88$. The full working table is shown below:
| Event | Prior | Likelihood | Unnorm | Posterior |
|---|---|---|---|---|
| Knows | 0.6 | 1.0 | 0.60 | 0.88 |
| Doesn’t know | 0.4 | 0.2 | 0.08 | 0.12 |