Conditional probability of an event with two conditions

Question

I am trying to solve the following problem:

In a certain test there are multiple choice questions. There are four possible answers to each question and one of them is correct. A student that studied solves 90% of the questions correctly by reasoning and for the remaining 10% he/she guesses. A student that did not study solves 20% of the questions correctly by reasoning and the remaining 80% by guessing.

A student that studied gets the correct answer, what is the probability that he/she was guessing?

I have been working to the solution like this:

P(study)=0.5
    P(guess|study)=0.5*0.1=0.05
        P(right|study n guess)=1/4*1/2*1/10=1/80
        P(wrong|study n guess)=3/4*1/2*1/10=3/80
    P(reason|study)=0.5*0.9=0.45
        P(right|study n reason)=1*1/2*9/10=9/20
        P(wrong|study n reason)=0*1/2*9/10=0
P(no_study)=0.5
    P(guess|no_study)=0.5*0.8=0.4
        P(right|no_study n guess)=1/4*1/2*8/10=8/80=1/10
        P(wrong|no_study n guess)=3/4*1/2*8/10=24/80=3/10
    P(reason|no_study)=0.5*0.2=0.1
        P(right|no_study n reason)=1*1/2*2/10=2/20=1/10
        P(wrong|no_study n reason)=0*1/2*2/10=0

The question is what is the probability that the student was guessing given that he/she is sudy and correct:

P(guess|study n right) = P(guess)P(study n right|guess) / P(guess)P(study n right|guess) + P(reason)P(study n right|reason)

To solve the equation I would need to know:

P(guess)
P(study n right|guess)
P(reason)
P(study n right|reason)

Would this be the way to calculate the probability of guessing and reasoning?

P(guess)=P(guess|study)+P(guess|no_study)=0.4+0.05=0.45

P(reason)=P(reason|study)+P(reason|no_study)=0.45+0.1=0.55

What I don't know how to calculate is:

P(study n right|guess)
P(study n right|reason)
```

Answer 1

You don’t need to know the probability that the student studies because it’s already given that he did. $$P(\text{guess | correct ans})$$ $$=\frac{P(\text{guess}) \cdot P(\text{correct ans | guess})}{P(\text{guess}) \cdot P(\text{correct ans | guess}) + P(\text{he knew the ans}) \cdot P(\text{correct ans | he knew the ans})}\\$$ $$= \frac{10\% \times \frac 14}{10\% \times \frac 14 + 90\% \times 1}\\$$ $$\\ = \frac{1}{37}$$