113 7.2. Bayes’ Formula — Mathematics for Public and Occupational Health Professionals
Bayes’ Formula
In this section, we will develop and use Bayes’ Formula to solve an important type of probability problem. Bayes’ formula is a method of calculating the conditional probability P(F | E) from P(E | F). The ideas involved here are not new, and most of these problems can be solved using a tree diagram. However, Bayes’ formula does provide us with a tool with which we can solve these problems without a tree diagram. We begin with an example.
Let JI be the event that Jar I is chosen, JII be the event that Jar II is chosen, B be the event that a black marble is chosen and W the event that a white marble is chosen. We illustrate using a tree diagram.
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This is a statement of Bayes’ formula. Bayes’ Formula:Let S be a sample space that is divided into n partitions, A1, A2, . . . An. If E is any event in S, then:
For certain problems, we can use a much more intuitive approach than Bayes’ Formula.
Practice questions1. Jar I contains five red and three white marbles, and Jar II contains four red and two white marbles. A jar is picked at random and a marble is drawn. Draw a tree diagram and find the following probabilities: 2. The table below summarizes the results of a diagnostic test:
Using the table, compute the following: 3. A computer company buys its chips from three different manufacturers. Manufacturer I provides 60% of the chips, of which 5% are known to be defective; Manufacturer II supplies 30% of the chips, of which 4% are defective; while the rest are supplied by Manufacturer III, of which 3% are defective. If a chip is chosen at random, find the following probabilities: 4. The following table shows the percent of “Conditional Passes” that different types of food premises received in a city during their last public health inspection.
If a premise is selected at random, find the following probabilities:
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