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Unformatted text preview: Stat 225 Lecture Notes “Conditional Probability” Ryan Martin Spring 2008 1 Motivation The conditional probability of an event is the probability that the event occurs given that some other event has occurred. As the following example illustrates, conditional probabilities can often be counter-intuitive. 1.1 Disease Detection A rare disease occurs in 0 . 1% of the population. Fortunately, there is a screening test which is correct with probability 0.99; i.e. if you have (do not have) the disease, then the test will detect (not detect) the disease 99% of the time. Suppose that Mike goes to get tested and the result is positive. How concerned should Mike be that he actually has the disease? To investigate this question, consider an imaginary population of size 100,000. According to the model, the population can be categorized as in the table below. Have the disease Do not have the disease Test positive 99 999 Test negative 1 98901 Exercise 1.1. Given that Mike tests positive, what is the probability that he actually has the disease? Is this surprising? 1.2 “Let’s Make a Deal!” This TV game show was popular during the 1960’s, hosted by Monty Hall. In one of the games on the show, a contestant was selected from the audience and asked to choose one of three doors. Behind each of the three doors was a prize, one of the prizes was valuable (say, a car) and the other two were “zonks” (say, goats), and the contestant would win the prize behind the selected door. Once the player chose a door, Monty would open one of the remaining doors hiding a zonk. The player would then be offered the opportunity to switch. What should the player do? Do you think there is a best strategy? Why? We will revisit this later. 1 2 Specifics 2.1 Definitions and Basic Properties Definition 2.1. Let A and B be events of a sample space with P ( A ) > 0. The conditional probability that B occurs given that A occurs is denoted P ( B | A ) and satisfies P ( B | A ) = P ( A ∩ B ) P ( A ) . Effectively, this means that we are reducing the sample space from Ω to A and P ( B | A ) is the “relative” probability of B . Draw a Venn Diagram. Exercise 2.2. What is the probability of... (a) drawing ♦ 2 given you draw from the full deck? (b) drawing ♦ Q given you draw from a deck of face-cards only? (c) drawing ♦ 2 given you draw from a deck of face-cards only? Exercise 2.3. Suppose three fair coins are tossed. Let A be the event that exactly two H ’s are tossed and B the event that the first toss is H . Find P ( B | A )....
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