Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.17

Chapter 5, Problem 5.2.17

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.

Verified step by step guidance

Step 1: Recognize that this is a binomial probability problem. The problem involves a fixed number of trials (n = 8), two possible outcomes (believe or do not believe), and a constant probability of success (p = 0.40). The random variable X represents the number of consumers who believe that cash will be obsolete in the next 20 years.

Step 2: Define the probability mass function (PMF) for a binomial distribution. The formula is: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient, p is the probability of success, and k is the number of successes.

Step 3: To find the probability that fewer than 3 consumers believe cash will be obsolete, calculate P(X < 3), which is the sum of probabilities for X = 0, X = 1, and X = 2. This can be expressed as: P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2).

Step 4: Compute each term using the binomial PMF formula. For example, P(X = 0) = (8 choose 0) * (0.40)^0 * (0.60)^8, P(X = 1) = (8 choose 1) * (0.40)^1 * (0.60)^7, and P(X = 2) = (8 choose 2) * (0.40)^2 * (0.60)^6.

Step 5: Add the computed probabilities from Step 4 to get the final result for P(X < 3). This sum represents 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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Key Concepts

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

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes. In this context, a 'success' is a consumer believing that cash will be obsolete, with a probability of 0.4. The distribution is defined by parameters n (number of trials) and p (probability of success), allowing us to calculate probabilities for different numbers of successes.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain threshold. In this question, we need to find the cumulative probability of fewer than 3 consumers believing in the obsolescence of cash, which involves summing the probabilities of 0, 1, and 2 successes in our binomial distribution.

Probability Mass Function (PMF)

The probability mass function (PMF) gives the probability of each possible outcome for a discrete random variable. For a binomial distribution, the PMF can be calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where k is the number of successes. This function is essential for determining the probabilities needed to answer the question about the selected consumers.

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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

In Exercises 25–28, find the probabilities and answer the questions.

Whitus v. Georgia In the classic legal case of Whitus v. Georgia, a jury pool of 90 people was supposed to be randomly selected from a population in which 27% were minorities. Among the 90 people selected, 7 were minorities. Find the probability of getting 7 or fewer minorities if the jury pool was randomly selected. Is the result of 7 minorities significantly low? What does the result suggest about the jury selection process?

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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

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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

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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Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Murders In a recent year (365 days), there were 650 murders in Chicago. Find the mean number of murders per day, then use that result to find the probability that in a single day, there are no murders. Would 0 murders in a single day be a significantly low number of murders?

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