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Chapter 4: Problem 45

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. In words, define the random variable \(X.\)

Short Answer

Expert verified
The random variable \(X\) is the number of freshmen asked until finding a 'yes' response.

Step by step solution

01

Identification of the Trial Type

Determine what type of probability distribution applies to the scenario. Since we are repeatedly selecting freshmen until we find the first 'yes', this scenario is a Bernoulli trial. The number of trials until the first success follows a geometric distribution.
02

Define Success in the Context

In this context, a 'success' is defined as finding a freshman who replies 'yes'. The probability of a freshman saying 'yes' is given as 71.3%, or 0.713.
03

Express the Random Variable in Words

The random variable \(X\) represents the number of freshmen that must be asked before encountering the first freshman who replies 'yes'. Geometrically, this is the waiting time until the first success.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bernoulli trial
A Bernoulli trial is a fundamental concept in probability and statistics. It refers to a random experiment where there are only two possible outcomes: success or failure.
In the context of the original exercise, asking a freshman whether they believe same-sex couples should have legal marital rights is a Bernoulli trial. Each individual response is independent of others, and each has a probability of a 'yes' response (success) or a 'no' response (failure).
  • There are only two potential outcomes.
  • Each trial is independent of the others.
  • The probability of getting a 'yes' remains constant for each specific question asked.
Understanding Bernoulli trials is crucial because it lays the groundwork for exploring how often these favorable outcomes occur under repeated conditions, which is vital for planning under uncertainty.
Probability Distribution
A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.
In this exercise, the probability distribution in question is a geometric distribution, which is associated with Bernoulli trials. A geometric distribution specifically models the number of trials needed to achieve the first success.
The probability of success on any given trial is defined, and this probability helps determine the shape of the distribution:
  • The probability of getting the first success on the first trial is highest, then it decreases with each subsequent trial.
  • This distribution is defined solely by the probability of success for each trial.
The mathematical form uses the probability of success to provide complete insights into how outcomes are distributed across trials.
Random Variable Definition
A random variable is a quantitative variable whose outcome is determined by the outcome of a random phenomenon.
For this exercise, the random variable, denoted as \(X\), represents how many freshmen you need to ask before hearing a 'yes' response.
This is a significant notion because it allows us to understand and measure uncertainty in practical terms.
  • \(X\) is a countable quantity indicating the number of attempts needed.
  • The variable can take any integer value starting from 1, depending on when the first 'yes' is heard.
  • This approach helps filter through randomness to find structured insights about likelihood in the context studied.
By using random variables, we can simulate real-world scenarios and build models to anticipate different outcomes.
Success Probability
Success probability is a critical component of Bernoulli trials and their resulting probability distributions.
In the scenario at hand, the success probability is defined as the likelihood that any given freshman will answer 'yes' to the survey question. Here, this probability is 0.713, or 71.3%.
The concept of success probability is central because it drives the expected value of the geometric distribution.
  • This probability remains constant across each independent trial.
  • Influences the shape of the distribution by impacting how quickly or slowly a success might occur.
  • Dictates how predictions about future events are made in the scenario.
An understanding of success probability allows for the application of statistical methods to predict outcomes efficiently.

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Most popular questions from this chapter

Find the standard deviation. $$\begin{array}{|c|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} & {(x-\mu)^{2} P(x)} \\ \hline 2 & {0.1} & {2(0.1)=0.2} & {(2-5.4)^{2}(0.1)=1.156} \\\ \hline 4 & {0.3} & {4(0.3)=1.2} & {(4-5.4)^{2}(0.3)=0.588} \\ \hline 6 & {0.4} & {6(0.4)=2.4} & {(6-5.4)^{2}(0.4)=0.144} \\ \hline 8 & {0.2} & {8(0.2)=1.6} & {(8-5.4)^{2}(0.2)=1.352} \\ \hline \end{array}$$

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. The chance of an IRS audit for a tax return with over \(\$ 25,000\) in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent. a. In words, define the random variable \( X\). b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ___ (___,____) d. How many audits are expected in a 20-year period? e. Find the probability that a person is not audited at all. f. Find the probability that a person is audited more than twice.

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ___ (___,____) d. How many are expected to not to use the foil as their main weapon? e. Find the probability that six do not use the foil as their main weapon. f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically.

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status. Construct the probability distribution function (PDF). $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline \\ \hline {} \\ \hline \\\ \hline \\ \hline {} \\ \hline \\ \hline {} \\ \hline \\ \hline \\\ \hline\end{array}$$

According to a recent article the average number of babies born with significant hearing loss (deafness) is approximately two per 1,000 babies in a healthy baby nursery. The number climbs to an average of 30 per 1,000 babies in an intensive care nursery. Suppose that 1,000 babies from healthy baby nurseries were randomly surveyed. Find the probability that exactly two babies were born deaf.
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