Question: You are an incumbent facing a challenger of an unknown quality. You are going to take part in several debates with the opponent. The challenger is equally likely to be a low, intermediate, or high quality politician, Depending the quality of the challenger, the chances of you winning an individual debate are 80%, 40%, or 20%, respectively (for this problem, assume that you either win or lose a debate).

1. What is the probability of you winning the first debate?
2. You've won the first debate! Given this information, what is the probability that you are facing a high quality politician as a challenger?
3. Extra Credit Again, keeping in mind that you've won the first debate, what is the probability that you will win the second debate? Assume that given the quality of the challenger, the debate outcomes are independent.
4. Explain the distinction between assuming that the outcomes of the debates are independent and assuming that they are conditionally independent given the quality of challenger. Which of these assumptions seems more reasonable, and why?

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