Elementary Statistics – Conditional Probability (Bayes’ Rule) Examples

Important Note: Although these problems illustrate Bayes’ Rule, you do not need to memorize the formula for the rule—in fact, I encourage you to solve each problem individually via a tree diagram.

 

  1. The probability of a certain disease is 0.01. A diagnostic test for the disease is developed. The test correctly diagnoses an infected person with probability 0.95. However, the test incorrectly diagnoses an uninfected patient with probability 0.06. If the test diagnoses a patient as having the disease, what is the conditional probability that the person really has the disease? Why is this probability so low?

 

 

 

 

 

 

 

 

 

  1. Daniel Kahneman (University of British Columbia) and Amos Tversky (Stanford) are two well-respected, widely published research psychologists. They have done much work on the biases and heuristics used in judgment and decision making.

 

Consider the following question, which they have given to many experimental subjects:

 

A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data:

(a)     85% of the cabs in the city are Green and 15% are Blue

(b)    a witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

 

What is the probability that the cab involved in the accident was Blue rather than Green?

 

The typical answer given to this question is 0.8, yet that answer is incorrect. What is the correct answer? And why do you think so many people get the question wrong?

 

 

 

 

 

 

 

 

  1. About 99% of babies born in the United States survive. Furthermore, Caesarian section is used in about 20% of births. Given a Caesarian section is used, about 96% of babies survive (note that while the previous two numbers are fairly accurate, this number is actually made up).

 

Given that a Caesarian section is not used, about what percent of babies survive?