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Chapter 6: Problem 16

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. According to Nielsen Media Research, \(70 \%\) of all U.S. households have cable television. In a small town of 40 households, a random sample of 10 households is asked whether they have cable television. The number of households with cable television is recorded.

Short Answer

Expert verified
The probability experiment represents a binomial experiment.

Step by step solution

01

- Identify Criteria for Binomial Experiment

A binomial experiment must meet the following criteria: 1) The experiment consists of a fixed number of trials. 2) Each trial is independent of the others. 3) There are only two possible outcomes (success or failure) in each trial. 4) The probability of success remains the same for each trial.
02

- Fixed Number of Trials

Determine if there is a fixed number of trials. In this case, the experiment consists of asking a random sample of 10 households out of 40. Thus, there are 10 trials.
03

- Independence of Trials

Check if each trial is independent. Each household is asked whether they have cable television individually, implying that the response of one household does not affect the response of another. Therefore, the trials are independent.
04

- Two Possible Outcomes

Verify if there are only two possible outcomes for each trial. In this case, each household either has cable television (success) or does not have cable television (failure). Therefore, there are two possible outcomes.
05

- Constant Probability of Success

Check if the probability of success remains constant across trials. According to Nielsen Media Research, the probability of a household having cable television is 70% or 0.70. This probability remains consistent for each household in the sample.
06

- Conclusion

Since the experiment meets all four criteria (fixed number of trials, independence of trials, only two possible outcomes, and a constant probability of success), it represents a binomial experiment.

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Key Concepts

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

Probability
In a binomial experiment, probability plays a crucial role. Probability is the chance that a particular event will happen. To calculate the probability, you divide the number of successful outcomes by the total number of possible outcomes.

In our example, the probability of a household having cable television is given as 70% or 0.70. This is the probability of 'success' for each trial. In mathematical terms, probability is denoted as P. Thus, in this case, the probability of success (P) is 0.70. This probability remains constant for each of the 10 households sampled, making it easy to compute the overall chance of getting a certain number of 'successes' in a fixed number of trials, which is essential for a binomial experiment.
Independence of Trials
For an experiment to be binomial, the trials must be independent. This means the outcome of one trial does not affect the outcome of another. Each trial is a separate event.

In our exercise, each household is asked independently about their cable television status. The response from one household does not influence the response of another household. If one household has cable, it does not change the probability of another household having cable. This independence is essential for calculating the binomial probability because it ensures that each trial is unique and unaffected by previous trials.
Fixed Number of Trials
A binomial experiment requires a fixed number of trials. This means that the number of times the experiment is conducted is decided beforehand and does not change.

In the provided exercise, the experiment involves selecting a sample of 10 households from a total of 40 and asking them about their cable television status. Here, the number of trials is 10, which is fixed. Having a fixed number of trials is crucial because it allows for the use of binomial formulas to calculate probabilities accurately. Without knowing the number of trials, it would be impossible to compute the binomial probabilities.
Constant Probability
For a binomial experiment, the probability of success must remain constant for each trial. This means that no matter how many trials you conduct, the probability of success in each trial is the same.

In our example, the probability that a household has cable television is 70% (or 0.70) for each of the 10 households sampled. This constant probability is vital because it simplifies the process of calculating the overall probability of getting a particular number of successes in a fixed number of trials. If the probability of success varied from trial to trial, it would no longer meet the criteria for a binomial experiment, and more complex calculations would be needed.

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Most popular questions from this chapter

Allergy Sufferers Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, \(5 \%\) of the patients in the study experienced insomnia as a side effect. (a) If 240 users of Clarinex-D are randomly selected, how many would we expect to experience insomnia as a side effect? (b) Would it be unusual to observe 20 patients experiencing insomnia as a side effect in 240 trials of the probability experiment? Why?

Suppose a life insurance company sells a \(\$ 250,000\) one-year term life insurance policy to a 20 -yearold male for \(\$ 350 .\) According to the National Vital Statistics Report, Vol. \(53,\) No. \(6,\) the probability that the male survives the year is \(0.998611 .\) Compute and interpret the expected value of this policy to the insurance company.

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. \(n=9, p=0.75\)

Some standardized tests, such as the SAT test, incorporate a penalty for wrong answers. For example, a multiple-choice question with five possible answers will have 1 point awarded for a correct answer and \(\frac{1}{4}\) deducted point for an incorrect answer. Questions left blank are worth 0 points. (a) Find the expected number of points received for a multiple-choice question with five possible answers when a student just guesses. (b) Explain why there is a deduction for wrong answers.

Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The number of defects in a roll of carpet. (b) The distance a baseball travels in the air after being hit. (c) The number of points scored during a basketball game. (d) The square footage of a house.
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