I'm quite inexperienced in mathematics so I apologize in advance for using incorrect terms or fundamentally misunderstanding the application of Bayes' Theorem.
My question is about whether you can apply Bayes' Theorem in the given scenario:
- A student is attempting to answer a question.
- The underlying probability that a student will get a question in this category wrong is $50\%$. This is based on past data of how all questions have been answered so far.
- A teacher thinks that the probability a student will get one specific question wrong is $70\%$. The teacher is accurate $90\%$ of the time.
- What is the overall probability the student will get this specific question wrong?
My current understanding is that Bayes' Theorem gives this relationship:
$$ \text{Q wrong} = \text{"Student will get this specific question wrong"} \\ \text{T's guess} = \text{"Teacher thinks they will get this question wrong"} \\ $$
$$ P(\text{Q wrong}|\text{T's guess}) = \frac{P(\text{T's guess}|\text{Q is wrong})P(\text{Q is wrong})}{P(\text{T's guess})} $$
The interpretation being:
- $P(\text{Q wrong})$ = Prior belief or "base rate"
- $P(\text{Q wrong}|\text{T's guess})$ = Posterior, updated belief based on teacher's estimate
However, this model doesn't seem right. It doesn't fit the normal examples of Bayes' Theorem I've seen, like a medical test with a false positive rate since the teacher isn't just stating that they will get the question wrong, instead giving an estimate of the probability.
Very un-mathematically blindly plugging the numbers in doesn't work unfortunately:
$$ \frac{0.7 \times 0.5}{0.9} \approx 0.389 $$
Which isn't correct because the probability should be above the baseline estimate.
Is it possible to answer this question using Bayes' Theorem or using some other method? Am I missing something crucial that makes this question impossible to ask without other information?
Thanks in advance.