A professor gives only two types of exams: easy and difficult. The probability of a difficult exam is 0.8. At the same time, if the exam is difficult, the first example is marked as challenging with a probability of 0.9, for an easy exam it is only 0.15.
a) What is the probability that the first question on your exam will be marked as challenging ?
b) If you find that the first question is marked challenging, what is the probability that you have a difficult exam?
My solution:
Probability of first question being challenging:
$\mathbb{P}=\frac{0.15*0.9}{0.8*0.9+0.2*0.15}$
Probability of the second case:
$\mathbb{P}=\frac{0.8*0.9}{0.8*0.9+0.2*0.15}$
Is that a correct solution?
Thanks
Best Answer
I agree to your solution of b). Let me define the events:
D: Difficult exam, $\overline D:$ Easy exam
C: First question is marked as challanging, $\overline C:$ First question is not marked as challanging
For a) you use the law of total probability:
$$P( C)=P(D\cap C)+P(\overline D\cap C),$$
where $P(D\cap C)=P(D)\cdot P(C|D)$ and $P(\overline D\cap C)=P(\overline D)\cdot P(C|\overline D)$
Can you proceed?