Question: Optional and a bit more challenging. Lets assume that in the problem above the percentage correct for the questions were as follows (assume each question is an independent event and students who answered the first question correctly had no advantage over students who did not:

\[\begin{aligned} & \text{Question 1 }\text{ 8}0\text{ }%\text{ correct} \\ & \text{Question 2 }\text{ 5}0%\text{ correct} \\ & \text{Question 3 }\text{ 9}0%\text{ correct} \\ \end{aligned}\]

What is the probability that a randomly chosen student will have gotten all three correct?

What is the probability that a randomly chosen student will have gotten all three incorrect?

What is the probability that a randomly chosen student will get the last two questions correct given that he got the first question incorrect? (Assume the likelihood of correctly answering any question is independent of performance on the other questions).