Watch the video on Setting up a Tree Diagram (Transcript) What is the probability that a randomly chosen athlete tests positive for banned substances? If two athletes are randomly selected, what's the probability that at least one of them tests positive?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Calculated Risks
Many high schools now have drug-testing programs for athletes. The main goal of these programs is to reduce the use of banned substances by students who play sports. It is not practical to test every athlete for drug use regularly. Instead, school administrators give drug tests to randomly selected student athletes at unannounced times during the school year. Students who test positive face serious consequences, including letters to their parents, required counseling, and suspension from athletic participation.
Drug test aren't perfect. Sometimes the tests say that athletes took a banned substance when they did not. This is known as a false positive. Other times, drug tests say that athletes are "clean" when they did take a banned substance. This is called a false negative.
Suppose that 16% of the high school athletes in a large school district have taken a banned substance. The drug test used by this district has a false positive rate of 5% and a false negative rate of 10%. If a randomly chosen athlete tests positive, what is the chance that the student actually took a banned substance. Use what you have learned in this chapter to help answer the following questions about the district's drug-testing program.
- Watch the video on Setting up a Tree Diagram (Transcript)
- What is the probability that a randomly chosen athlete tests positive for banned substances?
- If two athletes are randomly selected, what's the probability that at least one of them tests positive?
- What's the probability that a randomly selected athlete did not take a banned substance, given they tested positive? Based on your answer, do you think an athlete who tests positive should be suspended from athletic competition for a year? Why or why not?
- What's the probability that a randomly selected athlete took a banned substance given the student tested negative? Explain why it makes sense for th edrug-testing process to be designed so that this probability is less than the one you found in Question 4.
- The district decides to immediately retest and athlete who tests positive. Assume that the results of an athlete's two tests are independent. Find the probability that a student who gets a positive result on both tests actually took a banned substance (hint: took the banned substance given two positive tests). Based on your answer, do you think that an athlete who tests positive twice should be suspended from athletic competition for a year. Why or why not?
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4. If a randomly chosen athlete tests negative, what’s the probability that the student took a banned substance? Explain why it makes sense for the drug-testing process to be designed so that this probability is less than the one you found in Question 3.
5. The district decides to immediately retest any athlete who tests positive. Assume that the results of an athlete’s two tests are independent. Find the probability that a student who gets a positive result on both tests actually took a banned substance. Based on your answer, do you think that an athlete who tests positive twice should be suspended from athletic competition for a year? Why or why not?
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