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Chapter 5: Problem 15

False Positives The ELISA is a test to determine whether the HIV antibody is present. The test is \(99.5 \%\) effective, which means that the test will come back negative if the HIV antibody is not present \(99.5 \%\) of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is \(0.005 .\) Suppose that the ELISA is given to five randomly selected people who do not have the HIV antibody. (a) What is the probability that the ELISA comes back negative for all five people? (b) What is the probability that the ELISA comes back positive for at least one of the five people?

Short Answer

Expert verified
The probability of all five tests being negative is approximately 97.52%. The probability of at least one test being positive is approximately 2.48%.

Step by step solution

01

- Identify the probability of a negative test

Since the test is 99.5% effective, the probability of getting a negative result if the person does not have the HIV antibody is 0.995.
02

- Determine the probability for all five tests

The tests are independent, so the probability that all five tests come back negative is the product of the individual probabilities:
03

- Calculate the probability for all five tests

Calculate
04

- Identify the probability of a positive test

The probability that at least one test comes back positive is the complement of all tests being negative. Use the formula:
05

- Calculate the complement

Calculate

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Key Concepts

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

False Positives
A 'false positive' occurs when a test indicates the presence of a condition, like HIV antibodies, even though the condition is not actually present. This can be concerning, especially for serious conditions, as it could lead to unnecessary stress and further testing. The probability of a false positive is a key metric in evaluating the effectiveness of any medical test.
Independent Events
In probability, events are considered 'independent' if the outcome of one event does not affect the outcome of another. For example, if you test five individuals for HIV using the ELISA test and none of them have HIV, each test result is independent of the others. This means the probability of each test coming back negative is not influenced by the results of the other tests. If the test is 99.5% effective, the probability of getting a negative result for each person is 0.995.
Complement Rule
The 'complement rule' is a fundamental concept in probability that helps calculate the likelihood of an event not occurring. If the probability of an event A occurring is denoted as P(A), then the probability of event A *not* occurring is given by 1 - P(A). For example, if the probability that all five ELISA tests come back negative is 0.995^5, the probability that at least one comes back positive is 1 - 0.995^5.
ELISA Test Effectiveness
The ELISA (Enzyme-Linked Immunosorbent Assay) test is designed to detect the presence of HIV antibodies with a high level of accuracy. An effectiveness rate of 99.5% means the test correctly identifies individuals without HIV antibodies 99.5% of the time. However, this also implies a 0.5% chance of a false positive, as discussed earlier. Understanding the test's effectiveness helps in interpreting the results and planning further medical evaluations.
HIV Testing in Statistics
HIV testing often involves statistical methods to understand the accuracy and reliability of different tests. Using probability, one can evaluate the likelihood of different outcomes, such as false positives or negatives. The ELISA test's effectiveness and the rare occurrence of false positives can be better understood through statistical analysis, which is crucial for public health measures and for individuals undergoing testing.

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Most popular questions from this chapter

In 1991 , columnist Marilyn Vos Savant posted her reply to a reader's question. The question posed was in reference to one of the games played on the gameshow Let's Make a Deal hosted by Monty Hall. Her reply generated a tremendous amount of backlash, with many highly educated individuals angrily responding that she was clearly mistaken in her reasoning. (a) Using subjective probability, estimate the probability of winning if you switch. (b) Load the Let's Make a Deal applet at www.pearsonhighered.com/sullivanstats. Simulate the probability that you will win if you switch by going through the simulation at least 100 times. How does your simulated result compare to your answer to part (a)? (c) Research the Monty Hall Problem as well as the reply by Marilyn Vos Savant. How does the probability she gives compare to the two estimates you obtained? (d) Write a report detailing why Marilyn was correct. One approach is to use a random variable on a wheel similar to the one shown. On the wheel, the innermost ring indicates the door where the car is located, the middle ring indicates the door you selected, and the outer ring indicates the door(s) that Monty could show you. In the outer ring, green indicates you lose if you switch while purple indicates you win if you switch.

According to a Gallup Poll, about \(17 \%\) of adult Americans bet on professional sports. Census data indicate that \(48.4 \%\) of the adult population in the United States is male. (a) Assuming that betting is independent of gender, compute the probability that an American adult selected at random is male and bets on professional sports. (b) Using the result in part (a), compute the probability that an American adult selected at random is male or bets on professional sports. (c) The Gallup poll data indicated that \(10.6 \%\) of adults in the United States are males and bet on professional sports. What does this indicate about the assumption in part (a)? (d) How will the information in part (c) affect the probability you computed in part (b)?

Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly. (a) What is the probability that the committee is composed of all Democrats? (b) What is the probability that the committee is composed of all Republicans? (c) What is the probability that the committee is composed of three Democrats and four Republicans?

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the following contingency table. $$ \begin{array}{lrrrrr} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & \mathbf{1 3 2 9} \\ \hline \text { Less likely } & 22 & 6 & 22 & 16 & \mathbf{6 6} \\ \hline \begin{array}{l} \text { Neither more } \\ \text { nor less likely } \end{array} & 282 & 201 & 164 & 118 & \mathbf{7 6 5} \\ \hline \text { Total } & \mathbf{5 4 2} & \mathbf{5 3 6} & \mathbf{5 4 6} & \mathbf{5 3 6} & \mathbf{2 1 6 0} \end{array} $$ (a) What is the probability that a randomly selected individual is 35-44 years of age, given the individual is more likely to buy a product emphasized as "Made in America"? (b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in America," given the individual is \(35-44\) years of age? (c) Are 18 - to 34 -year-olds more likely to buy a product emphasized as "Made in America" than individuals in general?

According to the U.S. National Center for Health Statistics, \(0.15 \%\) of deaths in the United States are 25 - to 34-year-olds whose cause of death is cancer. In addition, \(1.71 \%\) of all those who die are \(25-34\) years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been \(25-34\) years old?
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