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Chapter 6: Problem 15

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. An investor randomly purchases 10 stocks listed on the New York Stock Exchange. Historically, the probability that a stock listed on the NYSE will increase in value over the course of a year is \(48 \% .\) The number of stocks that increase in value is recorded.

Short Answer

Expert verified
The experiment is a binomial experiment because it meets all the conditions.

Step by step solution

01

- Define a Binomial Experiment

A binomial experiment must satisfy the following four conditions: 1) The experiment consists of a fixed number of trials. 2) Each trial has only two possible outcomes (success or failure). 3) The probability of success is constant for each trial. 4) The trials are independent.
02

- Analyze the Given Experiment

Identify if the given experiment meets the criteria of a binomial experiment: The investor purchases 10 stocks (fixed number of trials). There are only two outcomes for each stock (increase in value or not). The probability that a stock will increase in value is 0.48 (constant probability). Each stock's performance is independent of the others.
03

- Verify the Conditions

Check each condition: 1) Fixed number of trials: Yes, there are 10 stocks. 2) Two possible outcomes: Yes, increase or not. 3) Constant probability: Yes, 0.48 for each stock. 4) Independent trials: Yes, assuming that the stock performances are independent.
04

- Conclusion

Since all conditions for a binomial experiment are satisfied, the given experiment is a binomial experiment.

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Key Concepts

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

Probability Experiments
A probability experiment is an action or process with uncertain outcomes that can be measured. It aims to observe and quantify occurrences under specified conditions. The unpredictable nature of the results is the key aspect of such experiments. For example,
  • Flipping a coin.
  • Rolling a die.
  • Selecting a random sample of people for a survey.
These are classical examples of probability experiments. Each of these actions does not have a predictable output, which is why they serve as excellent candidates for probability studies.
Conditions for Binomial Distribution
For an experiment to qualify as a binomial experiment, it must meet four specific conditions:

1. Fixed Number of Trials: The experiment must have a predetermined number of trials. For example, if we are testing whether a coin lands on heads 10 times, our fixed number of trials is 10.
2. Two Possible Outcomes: Each trial must have exactly two possible outcomes, such as success or failure, heads or tails, etc.
3. Constant Probability: The probability of success should be the same for each trial. For example, getting a head in a coin flip has a constant probability of 0.5 in a fair coin.
4. Independent Trials: The outcome of one trial should not affect the outcome of another. Each trial must be independent of every other trial.
The given exercise meets all these criteria, making it a binomial experiment.
Independent Trials
Independent trials are a fundamental condition in a binomial experiment. When trials are independent, the outcome of one trial does not affect the outcome of another trial. For instance, in our given exercise, if one stock increases in value, it does not affect whether another stock will increase. The independence of trials ensures that the probability calculations remain straightforward and accurate. This is crucial because it allows the use of binomial probability formulas accurately.
Fixed Number of Trials
A fixed number of trials means that the experimenter determines the number of trials before starting the experiment. This number does not change during the course of the experiment. In our example, the investor randomly purchases 10 stocks. Therefore, the number of trials is fixed at 10. This stability helps in the easy computation of the probability distribution and a clear understanding of the experiment's scope. From a mathematical standpoint, this is represented as 'n' in binomial distribution formulas, where 'n' is the fixed number of trials.

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Most popular questions from this chapter

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.

High-Speed Internet According to a report by the Commerce Department in the fall of \(2004,20 \%\) of U.S. households had some type of high-speed Internet connection. (a) Compute the mean and standard deviation of the random variable \(X,\) the number of U.S. households with a high-speed Internet connection in 100 households. (b) Interpret the mean. (c) Would it be unusual to observe 18 U.S. households that have a high-speed Internet connection in 100 households? Why?

(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. \(n=9, p=0.75\)

Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The number of lightbulbs that burn out in the next week in a room of with 20 bulbs. (b) The time it takes to fly from New York City to Los Angeles. (c) The number of hits to a Web site in a day. (d) The amount of snow in Toronto during the winter.

Probability Applet Load the binomial applet on your computer. (a) Set the probability of success, \(p,\) to 0.8 and the number of trials of the binomial experiment, \(n,\) to \(10 .\) Simulate shooting 10 free throws for \(N=1 .\) How many were made? (b) Set the probability of success to 0.8 and the number of trials of the binomial experiment to \(10 .\) Simulate shooting 10 free throws \(N=1000\) times. Use the results of the simulation to estimate the probability of making 10 out of 10 free throws. (c) Use the binomial probability formula to compute the probability of making 10 out of 10 free throws if the probability of success is \(0.8 .\) (d) Use the results of the simulation to estimate the probability of making at least 8 out of 10 free throws. (e) Use the binomial probability formula to compute the probability of making at least 8 out of 10 free throws. (f) Determine the mean number of free throws made for the 1000 repetitions of the experiment. Is it close to the expected value?
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