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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.2.39a
Chapter 3, Problem 3.2.39a

"39. Reliability of Testing A virus infects one in every 200 people. A test used to detect the virus in a person is positive 80% of the time when the person has the virus and 5% of the time when the person does not have the virus. (This 5% result is called a false positive.) Let A be the event ""the person is infected"" and B be the event ""the person tests positive.""
a. Using Bayes' Theorem, when a person tests positive, determine the probability that the person is infected."

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Step 1: Identify the given probabilities and events. Let P(A) represent the probability that a person is infected, which is 1/200 or 0.005. Let P(B|A) represent the probability of testing positive given the person is infected, which is 0.8. Let P(B|A') represent the probability of testing positive given the person is not infected (false positive rate), which is 0.05. Finally, P(A') is the probability that the person is not infected, which is 1 - P(A) = 0.995.
Step 2: Use the law of total probability to calculate P(B), the probability of testing positive. This is given by P(B) = P(B|A)P(A) + P(B|A')P(A'). Substitute the known values into this formula.
Step 3: Apply Bayes' Theorem to calculate P(A|B), the probability that the person is infected given they tested positive. Bayes' Theorem states: P(A|B) = [P(B|A)P(A)] / P(B). Substitute the values of P(B|A), P(A), and P(B) into this formula.
Step 4: Simplify the numerator of the Bayes' Theorem formula, which is P(B|A)P(A). Then, simplify the denominator P(B) using the values calculated in Step 2.
Step 5: Divide the simplified numerator by the simplified denominator to find P(A|B). This will give the probability that the person is infected given they tested positive.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bayes' Theorem

Bayes' Theorem is a mathematical formula used to update the probability of a hypothesis based on new evidence. It relates the conditional and marginal probabilities of random events. In this context, it helps calculate the probability of a person being infected given a positive test result, incorporating prior probabilities and the likelihood of the test results.
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Bayes' Theorem

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which represents the probability of event A occurring given that event B is true. Understanding this concept is crucial for applying Bayes' Theorem, as it allows us to assess how the test results influence the likelihood of infection.
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Conditional Probability Rule

False Positive Rate

The false positive rate is the probability that a test incorrectly indicates a positive result when the condition is not present. In this scenario, it is given as 5%, meaning that 5% of healthy individuals will test positive for the virus. This rate is essential for calculating the overall accuracy of the test and understanding its implications in the context of Bayes' Theorem.
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Introduction to Matched Pairs
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