Question

A professor only gives two types of exams, "easy" and "difficult". you will
get a difficult exam with a probability of 0.80. The probability that the first question
on the exam will be marked as difficult is 0.90 if the exam is difficult and 0.15 otherwise.
What is the probability that the first question on your exam is marked as difficult. How many
the probability that your exam is difficult to remember the first question on the exam is marked as
difficult?

Answer 1

By the total probability formula

$P\left( 1st\,\,difficult \right) =P\left( 1st\,\,difficult|difficult\,\,exam \right) P\left( difficult\,\,exam \right) +\\+P\left( 1st\,\,difficult|easy\,\,exam \right) P\left( easy\,\,exam \right) =\\=0.9\cdot 0.8+0.15\cdot \left( 1-0.8 \right) =0.75$

By Bayes formula

$P\left( difficult\,\,exam|1st\,\,difficult \right) =\frac{P\left( 1st\,\,difficult|difficult\,\,exam \right) P\left( difficult\,\,exam \right)}{P\left( 1st\,\,difficult \right)}=\\=\frac{0.9\cdot 0.8}{0.75}=0.96$