I am trying to find a way to calculate the probability that a user will pass a test at a certain certification authority, based on the user's history with answering questions using our service. Looking at a single question at a time (I know how to combine the probabilities), I have the following data:
pass
- The percentage of users who passed the test at the certification authority on their first attempt who answered the question correctly.fail
- he percentage of users who failed the test at the certification authority on their first attempt who answered the question correctly.correct
- if the current user answered the question correctly.
Given an example question:
$pass = 0.96$
$fail = 0.42$
If the user answered the question correctly, I believe the probability that he will pass (given that question as the only evidence), is something like:
$P(pass) = \frac{0.96}{(0.96 + 0.42)} = 0.697$
Am I correct? What if the user answered the question wrongly the first time?
On average, 82% of our candidates pass the certification on the first attempt. Of everyone who tries to take the certification, only 50% passes.
Edit: What about a second question, where 70% of candidates who pass the test on their first attempt answer correctly, but 80% of candidates who fail the test on their first attempt answer correctly? (meaning $pass = 0.7$ and $fail = 0.8$) Answering this question incorrectly will contribute to passing the test, even though it logically shouldn't.