Textbook Question
Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
11. Returning a rented movie after the due date and receiving a late fee
4. Probability
Multiplication Rule: Independent Events
6:19 minutesProblem 3.2.39b
Textbook Question
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."b. Using Bayes' Theorem, when a person tests negative, determine the probability that the person is not infected.
Verified step by step guidance1
Step 1: Understand the problem and identify the given probabilities. The problem involves Bayes' Theorem, which is used to calculate conditional probabilities. The given probabilities are: P(A) = 1/200 (probability of being infected), P(A') = 199/200 (probability of not being infected), P(B|A) = 0.8 (probability of testing positive given the person is infected), and P(B|A') = 0.05 (probability of testing positive given the person is not infected).
Step 2: Define the complement event. Since we are asked to find the probability that the person is not infected given they test negative, we need to calculate P(A'|B'). Here, A' is the event 'the person is not infected,' and B' is the event 'the person tests negative.'
Step 3: Use the complement rule to find P(B'|A) and P(B'|A'). The complement rule states that P(B') = 1 - P(B). Therefore, P(B'|A) = 1 - P(B|A) = 1 - 0.8 = 0.2, and P(B'|A') = 1 - P(B|A') = 1 - 0.05 = 0.95.
Step 4: Apply Bayes' Theorem to calculate P(A'|B'). Bayes' Theorem is given by: P(A'|B') = [P(B'|A') * P(A')] / P(B'). To calculate P(B'), use the law of total probability: P(B') = P(B'|A) * P(A) + P(B'|A') * P(A').
Step 5: Substitute the known values into the formulas. Substitute P(B'|A), P(B'|A'), P(A), and P(A') into the formula for P(B') to calculate its value. Then, substitute P(B'), P(B'|A'), and P(A') into Bayes' Theorem to find P(A'|B').
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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, allowing us to calculate the probability of an event given prior knowledge. In this context, it helps determine the likelihood of a person being uninfected after receiving a negative test result.
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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 conditional probability is crucial for interpreting test results and assessing the reliability of diagnostic tests.
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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 important for evaluating the accuracy of the test and understanding the implications of test results in the context of disease prevalence.
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