Probabilities #2

Exercise 1: Car Talk Puzzler: False Positives and Ano-cranial Inversions (ACI)

RAY: Here it is, the new and final Puzzler of this current Puzzler season. There's a rare disease (ACI) that's sweeping through your town. Of all the people who are exposed to it, 0.1 percent of the people actually contract the disease. There are no symptoms until the disease actually occurs. However, there's a diagnostic test that can detect the presence of the disease up to a year before it strikes. So you could actually seek treatment. You with me?

TOM: I'm with you!

RAY: You go to your doctor, and he administers the test. It comes out positive. You say, "I'm done for!" Then you get a little bit encouraged. You say, "Wait a minute, doc, is this test 100 percent accurate?" Your doctor responds, "Well, not really. It's 95 percent accurate." In other words, 5 percent of the people who take the test will test positive but they don't really have the disease. [Note: Assume that chances of testing negative if you DO have the disease are also 5%]

Here's the question: Given that you tested positive, what are the chances that you actually have the disease?

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Exercise 2: Play Tennis

You face two opponents, Andre Agassiz and Dan "The Man" Wiegmann (who does not quite play as well as Andre). Let's say your chances of winning against Dan are 0.5, while your chances of winning against Andre are 0.08. You must play them in alternating order but you are allowed to choose whether you would like to play Andre or Dan first. To be declared winner of the tournament you must win two games IN A ROW.

Here's the question: What is your best strategy? Do you have better chances playing D-A-D or A-D-A. Illustrate your conclusion with an argument based on probabilities.

last modified: 1/23/13