# Wendy’s Chances

1. What would Wendy’s chances of having colon cancer be if the other probabilities remained the same as in the original example, except that the probability that a person in the general population has colon cancer only 0.1 percent (or 0.001)?

2. What would Wendy’s chances of having colon cancer be if the other probabilities remained the same as in the original example, except that the probability that a person in the general population has colon cancer 1 percent (0.01)?

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3. What would Wendy’s chances of having colon cancer be if the other probabilities remained the same as in the original example, except that the conditional probability that the test is positive, given that the patient has colon cancer, is only 50 percent (or 0.5)?

4. What would Wendy’s chances of having colon cancer be if the other probabilities remained the same as in the original example, except that the conditional probability that the test is positive, given that the patient has colon cancer, is 99 percent (or 0.99)?

5. What would Wendy’s chances of having colon cancer be if the other probabilities remained the same as in the original example, except that the conditional probability that the test is positive, given that the patient does not have colon cancer, is 1 percent (or 0.01)?

6. What would Wendy’s chances of having colon cancer be if the other probabilities remained the same as in the original example, except that the conditional probability that the test is positive, given that the patient does not have colon cancer, is 10 percent (0.1)?

7. Chris tested positive for cocaine once in a random screening test. This test has a sensitivity and specificity of 95 percent, and 20 percent of the students in Chris’s school use cocaine. What is the probability that Chris really did use cocaine?

8. As in problem 7, 20 percent of the students in Chris’s school use cocaine, but this time Chris tests positive for cocaine on two independent tests, both of which have a sensitivity and specificity of 95 percent. Now what is the probability that Chris really did use cocaine?

9. In your neighborhood, 20 percent of the houses have high levels of radon gas in their basements, so you ask an expert to test your basement. An in- expensive test comes out positive in 80 percent of the basements that actu- ally have high levels of radon, but it also comes out positive in 10 percent of the basements that do not have high levels of radon. If this inexpensive test comes out positive in your basement, what is the probability that there is a high level of radon gas in your basement?

10. A more expensive test for radon is also more accurate. It comes out posi- tive in 99 percent of the basements that actually have high levels of radon. It also tests positive in 2 percent of the basements that do not high levels of radon. As in problem 7, 20 percent of the houses in your neighborhood have radon in their basement. If the expensive test comes out positive in your basement, what is the probability that there is a high level of radon gas in your basement?

11. Late last night a car ran into your neighbor and drove away. In your town, there are 500 cars, and 2 percent of them are Porsches. The only eyewitness to the hit-and-run says the car that hit your neighbor was a Porsche. Tested under similar conditions, the eyewitness mistakenly classifies cars of other makes as Porsches 10 percent of the time, and correctly classifies Porsches as such 80 percent of the time. What are the chances that the car that hit your neighbor really was a Porsche?

12. Late last night a dog bit your neighbor. In your town, there are 400 dogs, 95 percent of them are black Labrador retrievers, and the rest are pit bulls. The only eyewitness to the event, a veteran dog breeder, says that the dog who bit your neighbor was a pit bull. Tested under similar low-light con- ditions, the eyewitness mistakenly classifies black Labs as pit bulls only 2 percent of the time, and correctly classifies pit bulls as pit bulls 90 percent of the time. What are the chances that dog who bit your neighbor really was a pit bull?

13. In a certain school, the probability that a student reads the assigned pages before a lecture is 80 percent (or 0.8). If a student does the assigned read- ing in advance, then the probability that the student will understand the lecture is 90 percent (or 0.9). If a student does not do the assigned reading in advance, then the probability that the student will understand the lec- ture is 10 percent (or 0.1). What is the probability that a student did the reading in advance, given that she did understand the lecture? What is the probability that a student did not do the reading in advance, given that she did not understand the lecture?