I got a situation where I want to calculate the probability of a answer being correct if multiple student marked it as correct.
So, here is the situation. I have a multiple choice question with 3 options, only one of them is correct. We are given that the student is 80% of time correct.
p(option is correct ans|student selected that option) = 0.8
Then what will be the probability of the option to be correct answer if 2 student marked it as correct. Certainly it will be more than 0.8. But I am not able to find how to calculate.
i.e. p(option is correct ans| student s1 and s2 selected that option) = ?
Is the data sufficient for calculation of this probability.
Earlier I was modeling this as finding probability of cancer if two tests are positive, but then in that case we were given the prior probability of cancer. Here we don't have prior probability of answer being correct. Can I assume that it is 1/3. Also then in denominator there will be a joint probabilty of two student selecting the option, which I think is not independent. It is independent conditionally given the answer is correct, but not in general.
Please correct me if my above model of this problem is wrong and how can i calculate this.
Can this be generalised when we have N options and X out of Y student selected the option.