Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.18

Chapter 5, Problem 5.2.18

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 no more than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.

Verified step by step guidance

Step 1: Identify the type of probability distribution involved. Since the problem involves a fixed number of trials (8 consumers), each with two possible outcomes (believe or do not believe), and a constant probability of success (40% or 0.4), this is a binomial probability problem.

Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 8, the probability of success (p) is 0.4, and the number of successes (x) is no more than 3. This means we need to calculate P(X ≤ 3).

Step 3: Use the formula for the binomial probability mass function (PMF) to calculate individual probabilities for x = 0, x = 1, x = 2, and x = 3. The formula is: P(X = x) = (n choose x) * p^x * (1-p)^(n-x), where (n choose x) = n! / [x! * (n-x)!].

Step 4: Sum the probabilities for x = 0, x = 1, x = 2, and x = 3 to find P(X ≤ 3). This is expressed as: P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).

Step 5: If desired, use a statistical calculator or software (e.g., Excel, R, or a graphing calculator) to compute the cumulative probability P(X ≤ 3) directly using binomial distribution functions, which can simplify the calculation process.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 defined as a consumer believing that cash will be obsolete. The distribution is characterized by parameters n (number of trials) and p (probability of success), allowing us to calculate probabilities for different numbers of successes.

Probability Mass Function (PMF)

The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This function is essential for determining the likelihood of various outcomes, such as finding the probability that no more than 3 consumers believe cash will be obsolete.

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 case, to find the probability that no more than 3 consumers believe cash will be obsolete, we need to sum the probabilities of 0, 1, 2, and 3 successes. This concept is crucial for answering questions that involve ranges of outcomes rather than specific values.

Recommended video:

5:37

Introduction to Probability

Related Practice

Textbook Question

If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers you selected), while the remaining B objects are of the other type (such as lottery numbers you didn’t select), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and objects of type B is

In New Jersey’s Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probability of getting exactly four winning numbers with one ticket.

101

views

Textbook Question

Independent Events Again assume that when randomly selecting a speaking character in a movie, the probability of getting a female is 0.331, as in Exercise 1. If we want to find the probability of 20 females when 50 different speaking characters are randomly selected from a population of 1500 speaking characters, are the 50 selections independent? Using the 5% guideline for cumbersome calculations, can they be treated as being independent?

143

views

Textbook Question

Biometric Security In a USA Today survey of 510 people, 270 (or 53%) said that we should replace passwords with biometric security, such as fingerprints. Use the following probabilities related to determining whether the result of 270 is significantly high (assuming the true rate is 50%). Is 270 significantly high? What should be concluded about the claim that the majority of the population says that we should replace passwords with biometric security? Explain.

" style="" width="430">

130

views

Textbook Question

Random Variable The accompanying table lists probabilities for the corresponding numbers of unlicensed software packages when four software packages are randomly selected in China. What is the random variable, what are its possible values, and are its values numerical?

" style="" width="330">

146

views

Textbook Question

Discrete or Continuous? Is the random variable given in the table from Exercise 1 discrete or continuous? Explain.

215

views

Textbook Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.

184

views