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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 5.2.33
Chapter 5, Problem 5.2.33

Exercises 33 and 34 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder.


HIV It is estimated that in the United States, the proportion of people infected with the human immunodeficiency virus (HIV) is 0.00343. In tests for HIV, blood samples from 50 different people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?

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Step 1: Understand the problem. The goal is to calculate the probability that a combined blood sample from 50 people tests positive for HIV. A combined sample tests positive if at least one person in the group is infected with HIV. The proportion of people infected with HIV in the United States is given as 0.00343.
Step 2: Define the probability of an individual not being infected with HIV. If the probability of being infected is 0.00343, then the probability of not being infected is calculated as 1 - 0.00343. This represents the complement of the infection probability.
Step 3: Calculate the probability that all 50 individuals in the combined sample are not infected. Since the infection status of each individual is independent, the probability that all 50 individuals are not infected is the product of the individual probabilities of not being infected. This can be expressed mathematically as \( P(\text{all not infected}) = (1 - 0.00343)^{50} \).
Step 4: Determine the probability that at least one person in the combined sample is infected. This is the complement of the probability that all 50 individuals are not infected. Mathematically, \( P(\text{at least one infected}) = 1 - P(\text{all not infected}) \).
Step 5: Interpret the result. Once the probability is calculated, compare it to a threshold (e.g., 0.05 for unlikely events) to determine whether it is unlikely for the combined sample to test positive. If the probability is very small, it suggests that it is unlikely for the combined sample to test positive.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps determine the chance that at least one individual in a combined sample of 50 people is infected with HIV. Understanding how to calculate probabilities, especially in composite scenarios, is crucial for interpreting the results of the combined blood tests.
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Binomial Distribution

The binomial distribution is a statistical distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, it can be used to calculate the probability of having at least one positive HIV test in a sample of 50 people, given the known infection rate. This concept is essential for understanding how to apply statistical methods to real-world testing scenarios.
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Composite Sampling

Composite sampling is a technique where multiple samples are combined into one for testing, which can reduce costs and time in laboratory settings. This method is particularly useful in medical testing, as it allows for efficient screening of large populations. Understanding composite sampling is key to grasping how the combined test results can indicate the presence of a disease among individuals in the sample.
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Related Practice
Textbook Question

Texting and Driving. In Exercises 21–26, refer to the accompanying table, which describes probabilities for groups of five drivers. The random variable x is the number of drivers in a group who say that they text while driving (based on data from an Arity survey of drivers).

Using Probabilities for Significant Events


a. Find the probability of getting exactly 3 drivers who say that they text while driving.


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Textbook Question

40% of consumers believe that cash will be obsolete in the next 20 years (based on a survey by J.P. Morgan Chase). In each of Exercises 15–20, assume that 8 consumers are randomly selected. Find the indicated probability.


Find the probability that at least 6 of the selected consumers believe that cash will be obsolete in the next 20 years.

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Textbook Question

40% of consumers believe that cash will be obsolete in the next 20 years (based on a survey by J.P. Morgan Chase). In each of Exercises 15–20, assume that 8 consumers are randomly selected. Find the indicated probability.


Find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.

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Textbook Question

Texting and Driving. In Exercises 21–26, refer to the accompanying table, which describes probabilities for groups of five drivers. The random variable x is the number of drivers in a group who say that they text while driving (based on data from an Arity survey of drivers).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 is a significantly high number of drivers who say that they text while driving.

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Textbook Question

Identifying Discrete and Continuous Random Variables. In Exercises 5 and 6, refer to the given values, then identify which of the following is most appropriate: discrete random variable, continuous random variable, or not a random variable.




a. IQ scores of statistics students

b. Exact heights of statistics students

c. Shoe sizes (such as 8 or 8½) of statistics students

d. Majors (such as history) of statistics students

e. The number of rolls of a die required for a statistics student to get the number 4

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Textbook Question

In Exercises 5–12, determine whether the given procedure results in a binomial distribution or a distribution that can be treated as binomial (by applying the 5% guideline for cumbersome calculations). For those that are not binomial and cannot be treated as binomial, identify at least one requirement that is not satisfied.


In a Pew Research Center survey, 3930 subjects were asked if they have ever fired a gun, and the responses consist of “yes” or “no.”

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