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Chapter 5: Problem 97

An athlete suspected of using steroids is given two tests that operate independently of each other. Test A has probability 0.9 of being positive if steroids have been used. Test \(\mathrm{B}\) has probability 0.8 of being positive if steroids have been used. What is the probability that neither test is positive if steroids have been used? (a) 0.72 (b) 0.38 (c) 0.02 (d) 0.28 (e) 0.08

Short Answer

Expert verified
(d) 0.28

Step by step solution

01

Understand the problem

We need to calculate the probability that neither test A nor test B is positive, given that steroids have been used. Both tests operate independently.
02

Identify the given probabilities

From the problem, we know: - Probability Test A is positive if steroids have been used: \( P(A^+) = 0.9 \)- Probability Test B is positive if steroids have been used: \( P(B^+) = 0.8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In the world of probability, independent events play a crucial role. These are events where the occurrence of one doesn't affect the occurrence of another. Imagine flipping a coin and rolling a die. The outcome of the coin flip doesn't influence the result of the die roll; these two actions are independent events.

In the context of our original exercise, we have two tests: Test A and Test B. These tests are designed to check for the use of steroids independently. This independence implies that the result of Test A does not impact the result of Test B. If test A comes out positive, it doesn鈥檛 mean test B has to come out positive as well, or vice-versa.

Understanding independence can simplify probability calculations greatly. When two events are independent, the combined probability of both events occurring is simply the product of their individual probabilities.
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred. It's like knowing that it's cloudy, and then wondering about the probability of rain. The cloudiness changes what we expect.

In our steroid test problem, we aren't dealing directly with conditional probability, because the tests are independent. However, the initial condition we have is that steroids have been used, which affects the probability of the tests being positive. This condition allows us to solely focus on the given probabilities for each test being positive when steroids are used.
Probability Calculation
Calculating probability for events involves different methods depending on the relationship between the events. When events are independent, like our tests here, probability calculations become more straightforward.

To find the probability that neither test A nor test B is positive when steroids have been used, we first determine the probability that each test individually comes out negative. For Test A, if the positivity probability is 0.9, then the probability of a negative result is \[P(A^-) = 1 - P(A^+) = 1 - 0.9 = 0.1\] For Test B, similarly, the probability of a negative result is \[P(B^-) = 1 - P(B^+) = 1 - 0.8 = 0.2\] Because the tests are independent, the probability that both are negative is simply the product of their individual negative probabilities:\[P(A^- \cap B^-) = P(A^-) \times P(B^-) = 0.1 \times 0.2 = 0.02\]Thus, the probability that neither test is positive if steroids have been used is 0.02.

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Most popular questions from this chapter

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