Probability Question

#1
Hi. I solved this question but I’m not sure that it is true. I don’t know the answer btw.(I couldn’t load the my solution but my answer is 17/30)

The Statistics exam is going to be difficult, normal, or easy. The probability that I solve a question on a difficult exam is 30%, on a normal exam it is 50%, and on an easy exam is 90%. These probabilities are valid for each of the questions on the exams, independent from one another. You take the exam and solve the first question. What is the probability that you will also solve the second question?
 
#2
Your answer appears to be incorrect, assuming that it is equiprobable that one is faced with one of an easy, normal, or difficult exam.

Conditional probability: P(A & B) = P(B) × P(B | A)

Let A = "Answers first question correctly" and B = "Answers second question correctly."

On the aforesaid assumption, you can calculate the overall probability of answering the second question correctly (or any one question, for that matter; use a tree diagram if you really have to). That would be the value P(B). You can also calculate the overall probability of answering the first two questions correctly (or any two questions, for that matter). That would be the value of P(A & B). You can then calculate P(B | A) directly from the above conditional probability formula.
 
#3
I didn’t understand how our answers are different because in your solution if the exam is difficult P(B|A)=3/10, if the exam is normal P(B|A)=1/2, and if the exam is easy P(B|A)=9/10. Then shouldn’t we add them together and multiply by 1/3?
 
#4
The values you assign to P(B | A) for the different exam situations are all incorrect, suggesting that you don't properly understand the problem.