Math115a Bayes’ Theorem #1 Name:________________________ReyesFor each problem on the worksheet: Define the random variables used. Use the correct mathematical notation. Use a tree diagram, Venn diagram, table or other method to show all the probabilities given in the problem.1. A test is designed to detect cancer. If a person has cancer, the probability that the test will detect the cancer is 0.95. If the person does not have cancer, then the probability that the test will erroneously indicate that he or she does have cancer if 0.1. If 3% of the population who will take the test have cancer, what is the probability that a person described by the test as having cancer does not really have it?2. Suppose there is a certain disease randomly found in one-half of one percent (.005) of the general population. A certain clinical blood test is 99 percent (.99) effective in detecting the presence of this disease; that is, it will yield an accurate positive result in 99 percent of the cases where the disease is actually present. But it also yields false-positive results in 5 percent (.05) of the cases where the disease is not present. Let A be the event that the disease is present in any particular person. Let B be the even that the test will yield a positive test result. What is the probability that a positive test result will be a true positive?3. The duration of the traffic light colors at a major intersection are as follows: green – 90 seconds; yellow – 4 seconds; and red – 40 seconds. A car has just passed legally through the intersection (either a green or yellow light was showing). Find the probability that a green light was showing.4. In a major city, 70% of the drivers are over 25 years old, and 12% of them will have a traffic violation during a 12-month period. The drivers 25 years and under comprise 30% of the drivers, and28% of them will have a traffic violation in a 12-month period. A driver is charged with a traffic violation. Find the probability that the driver is over
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