Section 1: Bayes' Theorem and Conditional Probability (22 points) and want to know how to interpret the results. From the CDC we know that 12% of adults smoke (cigarettes) regularly, another 8% smoke a few times a month or less (occasional smokers), and 80% are non-smokers. The test the company uses has the following historic conditional probabilities: 95% positive results among regular smokers, 20% positive results among occasional smokers, 5% positive results among non-smokers. The company plans to charge higher he rates to regular smokers, occasional smokers and non-smokers will be charged the same (lower) rates. 1. (4 points) Describing possible outcomes. - Create a two-way (contingency) table that shows the probability of the 6 distinct joint outcomes (type of smoker and test outcome). The "insides" of table should be joint probabilities (intersections) and you should also show the total (marginal) probabilities for each category. Place the smoking categories in the rows, and test result in the columns. A probability tree might help you organize the information. - Based on this, assuming those being tested are a random sample of the population, what percentage of tests should the company expect to come back positive? 2. (6 points) Assessing test results. - Create a new two-way table that shows the conditional probabilities of smoker type given a certain test result. In other words: what percentage of those testing positive should the company expect are regular smokers? Occasional smokers? Non-smokers? What percentage of those testing negative are in each category? - Create a chart (or set of charts) in Excel that displays the probability distributions across the test results. - Discuss these results. How confident should the company be in charging different rates based on the test results? Assume the company only cares about regular smokers in terms of prices