A professor gives only two types of exams - Math

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Q3 A professor gives only two types of exams, "easy" and “hard”. You will get a hard 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 hard and is 0.15 otherwise.

(a) Prove that the probability of the first question on your exam is marked as difficult is 0.735.

(b) Given that the first question on the exam is marked as difficult, show that there is a 98% chance that the exam is hard.

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Q3.

We define events as follows.

$\tiny E$ denotes the event that an examination is easy.
$\tiny H$ denotes the event that an examination is hard.
$\tiny D$ denotes the event that the first question on the examination is marked as difficult.