Question

A market research firm hires operators to conduct telephone surveys. The computer randomly dials a telephone number, and the operator asks the respondent whether or not he has time to answer some questions. Let \(x\) be the number of telephone calls made until the first respondent is willing to answer the operator's questions. Is this a binomial experiment? Explain.

Short answer

Explain your answer. Answer: No, this situation is not a binomial experiment because there is no fixed number of trials. Although the trial results are independent and the probability of success is constant, a fixed number of trials is a required condition for a binomial experiment, which is not present in this scenario.

Step-by-step solution

Identify the fixed number of trials

In this case, there is not a fixed number of trials as the number of calls made until a respondent is willing to answer questions (\(x\)) can vary. It can be any positive integer value of \(x\).

Check if the trial results are independent

For each call made by the operator, the outcome (whether the person answers the questions or not) should be independent of the other calls, as the computer is randomly dialing the numbers. Therefore, the results of the trial are independent.

Check if the probability of success is constant

The probability of respondents willing to answer the questions should be constant during the experiment, assuming the response rate doesn't change throughout the calls the operator makes.

Conclude if it is a binomial experiment or not

Although steps 2 and 3 satisfy the conditions for a binomial experiment, step 1 does not, as there is no fixed number of trials. Therefore, this is not a binomial experiment.

Key concepts

Random Variables

In probability theory, a random variable is a function that assigns a numerical value to each outcome of a random experiment. It's a way to quantify outcomes of random processes, like the number of heads in coin tosses or the roll of a die. Random variables can be:

• Discrete: Taking a countable number of distinct values, such as integers (e.g., number of customer arrivals).
• Continuous: Taking an infinite number of possible values, often within a range (e.g., heights, weights).

Given our exercise, "number of telephone calls made until the first respondent answers" is a discrete random variable. Each dial represents a possible outcome—either someone answers, or they don’t. Understanding that a random variable captures this variability is crucial for analyzing data from random experiments.

Binomial Experiment

A binomial experiment is a specific type of random experiment that meets a set of criteria. It involves exactly two possible outcomes for each trial, typically labeled as "success" and "failure." The criteria include:

• A fixed number of trials.
• Independent trials—where the result of one does not affect the others.
• A constant probability of success across all trials.

In our scenario, we checked these three conditions to determine if calling respondents could be modeled as a binomial experiment. While trials are independent, as each call's outcome has no effect on the next, and the probability of success could remain constant, the number of trials is not fixed. Thus, our scenario does not neatly fit the binomial framework, primarily because of the variable number of calls made until someone agrees to answer.

Independent Trials

In probability theory, trials are independent if the outcome of one trial does not influence the outcome of another. This independence is crucial because it validates using certain statistical methods and models. When conducting the telephone survey, each call is independent. The computer dials numbers randomly, making each call's outcome convenient and disconnected from others:

• The operator's success on call one doesn't increase or decrease success chances on the second call.
• Independence permits a straightforward calculation of probabilities across trials.

This independence forms one of the three foundations for a binomial experiment, increasing its usability in various modeling situations despite our scenario having only two of these necessary conditions.

Probability of Success

The probability of success is the likelihood of a desired outcome occurring in any given trial of an experiment. It is a fundamental concept in probability theory, often denoted by "p." For a binomial experiment to occur, this probability must remain constant across all trials. In the scenario of telephone surveying, if we assume that the conditions remain constant (e.g., time of day, type of respondents), we could claim that the probability of a respondent agreeing to answer questions would be consistent. This implies that:

• If 30% of people usually respond, we expect each call to have a 0.3 probability of success.
• This consistent figure simplifies calculating expected outcomes across multiple trials.

Therefore, even if the number of trials isn’t fixed in our experiment, maintaining a constant probability of success is still valuable for modeling other statistical features of the problem.