1. Explain how Bayes' theorem describes the process of updating one's beliefs based on new information. 2. Under what conditions can you calculate probabilities by counting outcomes? What axiom is responsible for this? 3. A diagnostic test is "perfect" if it has both 100% specificity and 100% sensitivity. Explain why a perfect test is not possible if the disease outcome we are testing for is random (i.e. not perfectly predictable). 4. What is the difference between independent events and disjoint events? 5. In order to fully specify a random variable, what two things do you need to write down? 6. If you want to completely write down the distribution of a random variable, what functions can you write down to do it? Discuss all of the ones we have encountered in this class. 7. If two random variables have the same mean and variance, does that mean they have the same distribution?