Worksheet 14: Common mistakes in conditional probability#
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You get tested for a rare disease affecting 1% of the population, with a test that is 99% accurate. When the test comes back positive, you’re certain that you have the disease. Why is this an example of the base rate fallacy?
In an experiment, two groups of people were given a description of a guy named Jack, along with some personality traits that are associated with engineers. Group 1 was told that Jack was sampled from a group of 100 men, 30% of whom were engineers and 70% of whom were lawyers. Group 2 was told that Jack was sampled from a group of 70% engineers and 30% lawyers. In both groups, participants were asked to give the probability of Jack being an engineer, based on his description. There was effectively no difference between group 1 and 2 in the median probability assigned to the event that Jack is an engineer. Why is this an example of people committing the base rate fallacy?
You are told that the state home to the second-largest number of NFL players is California; 151 out of 1696 players are from California (about 9%). Does being from California improve your chances of playing in the NFL? Phrase this question in the language of conditional probabilities.
Of 114 Scottish hiking fatalities in the last decade, 104 were men. On “wait wait don’t tell me” they say that the thing that increases your chance of dying the most while hiking in Scotland is being a man. What is wrong with this reasoning? Phrase it in the language of conditional probability.
In the Sally Clark case, what does the 1/73 million calculation assume? What’s wrong with this assumption?
What is the problem with the prosecution’s logic in the Sally Clark case?
\(25\%\) of erectile dysfunction patients are under 40.
a. Is this a large number? What is a natural “base rate” to compare this to?
b. Can you estimate \(\Pr[Y]\), where \(Y\) is the event of being under 40?
“Only 1 in 10 students fail their qualifying exam, so I don’t need to study hard for it.” Why is this an example of the defense attorney’s fallacy? Put it in the language of conditional probabilities.