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.q.7

Chapter 5, Problem 5.q.7

In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).

Probability Find the probability that at least one of the subjects is a sleepwalker.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the probability that at least one of the subjects is a sleepwalker. This can be calculated using the complement rule, which states that the probability of 'at least one' is equal to 1 minus the probability of 'none'.

Step 2: Identify the probability of 'none' from the table. The table provides the probability P(x) for different values of x, where x represents the number of sleepwalkers. For x = 0 (no sleepwalkers), P(0) = 0.172.

Step 3: Apply the complement rule. The formula for the probability of at least one sleepwalker is: P(at least one) = 1 - P(none). Substitute the value of P(none) from the table into this formula.

Step 4: Perform the subtraction. Subtract the probability of 'none' (P(0) = 0.172) from 1 to find the probability of at least one sleepwalker.

Step 5: Interpret the result. The resulting probability represents the likelihood that at least one of the five subjects is a sleepwalker based on the given data.

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.

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 quantifies the chance of sleepwalking occurrences among a group of five adults. The sum of probabilities for all possible outcomes must equal 1, allowing for the calculation of probabilities for specific events, such as at least one sleepwalker.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. For this question, to find the probability that at least one subject is a sleepwalker, one can calculate the probability that none are sleepwalkers and subtract it from 1. This simplifies the calculation and provides a clearer understanding of the event's likelihood.

Discrete Probability Distribution

A discrete probability distribution describes the probabilities of the possible values of a discrete random variable. In the provided table, the variable x represents the number of sleepwalkers in a group of five, and P(x) gives the probability of each outcome. Understanding this distribution is essential for analyzing the likelihood of different scenarios, such as determining the probability of at least one sleepwalker.

Recommended video:

Guided course

04:48

Variance & Standard Deviation of Discrete Random Variables

Related Practice

Textbook Question

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

Hurricanes

b. In a 118-year period, how many years are expected to have no hurricanes?

views

Textbook Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay $1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect $5000.

e. If you bet \$1 in North Carolina’s Pick 3 game, the expected value is Which bet is better in the sense of a producing a higher expected value: A \$1 bet in the North Carolina Pick 4 game or a \$1 bet in the North Carolina Pick 3 game?

views

Textbook Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.

e. What do the results suggest about how the clerk met the requirement of using a random method to assign the order of candidates’ names on voting ballots?

views

Textbook Question

Using Probabilities for Significant Events

d. Is 1 a significantly low number of matches? Why or why not?

Textbook Question

Hurricanes

a. Find the probability that in a year, there will be no hurricanes.

views

Textbook Question

In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).

Does the table describe a probability distribution?