Skip to main content
Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.R.21b
Chapter 4, Problem 4.R.21b

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
Fourteen percent of noninstitutionalized U.S. adults smoke cigarettes. After randomly selecting ten noninstitutionalized U.S. adults, you ask them whether they smoke cigarettes. Find the probability that the first adult who smokes cigarettes is (b) the fourth or fifth person selected.

Verified step by step guidance
1
Step 1: Identify the appropriate probability distribution for the problem. Since we are looking for the probability that the first adult who smokes cigarettes is the fourth or fifth person selected, this is a geometric distribution problem. The geometric distribution models the probability of the first success occurring on a specific trial.
Step 2: Write the formula for the geometric distribution. The probability that the first success occurs on the nth trial is given by: P(X = n) = (1 - p)^(n-1) * p, where p is the probability of success (in this case, the probability that an adult smokes cigarettes), and (1 - p) is the probability of failure.
Step 3: Calculate the probability for the fourth person being the first smoker. Substitute n = 4 and p = 0.14 into the formula: P(X = 4) = (1 - 0.14)^(4-1) * 0.14. Simplify the expression to find the probability for the fourth person.
Step 4: Calculate the probability for the fifth person being the first smoker. Similarly, substitute n = 5 and p = 0.14 into the formula: P(X = 5) = (1 - 0.14)^(5-1) * 0.14. Simplify the expression to find the probability for the fifth person.
Step 5: Add the probabilities from Step 3 and Step 4 to find the total probability that the first smoker is either the fourth or fifth person. Finally, compare the result to a threshold (e.g., 0.05) to determine whether the event is unusual.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. In this context, it applies to the scenario of finding the first adult who smokes cigarettes among a sample. The probability of success (smoking) is constant, and the trials continue until the first success occurs.
Recommended video:
Guided course
06:38
Intro to Frequency Distributions

Probability Calculation

Calculating probabilities involves determining the likelihood of a specific event occurring. For the geometric distribution, the probability of the first success occurring on the k-th trial can be calculated using the formula P(X = k) = (1-p)^(k-1) * p, where p is the probability of success. This formula helps in finding the probability that the first smoker is the fourth or fifth person selected.
Recommended video:
Guided course
07:09
Probability From Given Z-Scores - TI-84 (CE) Calculator

Unusual Events

An event is considered unusual if its probability is less than 0.05 (5%). In the context of this problem, after calculating the probabilities for the fourth and fifth adults being the first smokers, one must assess whether these probabilities fall below this threshold to determine if the events are unusual. This concept helps in interpreting the significance of the results.
Recommended video:
05:54
Probability of Multiple Independent Events
Related Practice
Textbook Question

In Exercises 7 and 8, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.


The number of cell phones per household in a small town

99
views
Textbook Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Fourteen percent of noninstitutionalized U.S. adults smoke cigarettes. After randomly selecting ten noninstitutionalized U.S. adults, you ask them whether they smoke cigarettes. Find the probability that the first adult who smokes cigarettes is (a) the third person selected.

105
views
Textbook Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities

Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (a) exactly four

109
views
Textbook Question

In Exercises 11 and 12, determine whether the experiment is a binomial experiment. If it is, identify a success; specify the values of n, p, and q; and list the possible values of the random variable x. If it is not a binomial experiment, explain why.


A fair coin is tossed repeatedly until 15 heads are obtained. The random variable x counts the number of tosses.

130
views
Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (b) at least three

105
views
Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Seventy-two percent of U.S. civilian employees have access to medical care benefits. You randomly select nine civilian employees. Find the probability that the number who have access to medical care benefits is (c) more than six.

74
views