What Are Examples Of Variables That Follow A Binomial Probab

What Are Examples Of Variables That Follow A Binomial Probability Dist

What are examples of variables that follow a binomial probability distribution? What are examples of variables that follow a Poisson distribution? When might you use a geometric probability? All calculations should be done in Word using Equation Editor and all charts and graphs should be done in Excel.

In the judicial case of the United States vs. the City of Chicago, discrimination was charged in a qualifying exam for the position of Fire Captain. The table below shows the number of candidates passed or failed in two groups: Group A (minority) and Group B (majority). Using this data, the following questions are addressed:

• a) If one of the test subjects is randomly selected, find the probability of getting someone who passed the exam.
• b) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group B or passed.
• c) Find the probability of randomly selecting two different test subjects and finding that they are both in Group A.
• d) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group A and passed the exam.
• e) Find the probability of getting someone who passed, given that the selected person is in Group A.

In a different context, about 35% of the population has blue eyes. For randomly selected individuals, the probability that a person does not have blue eyes can be determined using binomial probability principles. If four people are randomly selected, the probability that all four have blue eyes can be computed. Analyzing whether it would be unusual to observe this event involves comparing the probability to common significance thresholds.

For investment analysis, suppose the expected returns of two stocks under different economic conditions are distributed with certain probabilities. Calculations of expected values, standard deviations, and covariances help determine which stock presents a better investment opportunity. This involves applying probability distributions and statistical measures to compare risk-adjusted returns.

In the South Carolina Palmetto Cash 5 lottery game, players select five numbers between 1 and 38. Computing the total number of combinations (using combinatorial calculations), the probability of winning the jackpot, and assessing whether winning is unusual involves basic probabilistic and combinatorial reasoning.

Regarding market analysis, Sun Microsystems holds a 50% market share for high-end Unix machines. With 15 corporations purchasing such machines, the binomial distribution models the number of units sold by Sun Microsystems. Calculating expected sales, probabilities of specific purchase counts, and cumulative probabilities assists in market estimations.

Finally, in Dutchess County, NY, motor vehicle death counts are modeled using the Poisson distribution. Calculations involve determining the mean number of daily deaths, probabilities of more than two deaths in a day, and the probability of no deaths in a day.

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The variables that follow a binomial probability distribution are typically dichotomous, representing "success" or "failure," with fixed probability over fixed independent trials. Examples include the number of students passing an exam, the number of defective items in a batch, or the number of customers arriving at a store within an hour. These variables are characterized by parameters n (number of trials) and p (probability of success).

The Poisson distribution models the number of events occurring within a fixed interval or space when the events happen independently and at a constant average rate. Examples include the number of phone calls received at a call center per hour, the number of decay events from a radioactive source per minute, or the number of car accidents at an intersection in a day. The Poisson distribution is especially useful in modeling rare events over continuous intervals.

On the other hand, geometric probability pertains to the number of trials needed for the first success in a sequence of independent Bernoulli trials. An example is determining how many coin tosses are expected until the first head appears, or the number of customer calls until the first purchase. Geometric probability is useful for modeling scenarios where the interest is in the number of attempts before success.

Applying these distributions depends on the specific scenario. For example, in the Chicago fire captain exam case, the binomial distribution is appropriate because they involve fixed numbers of independent candidates, each with the same probability of passing. For the eye color example, the binomial distribution models the probability of a certain number of blue-eyed people in a sample. The investment scenario uses the binomial distribution to predict the number of favorable economic conditions, affecting returns. The lottery problem involves calculating the probability of a single event with a very low probability, suitable for combinatorial methods. The market share analysis applies the binomial distribution to model discrete units sold, while the motor vehicle deaths follow a Poisson distribution because they involve counting rare, independent events over time.

In practice, understanding which distribution applies allows organizations and individuals to make informed decisions, evaluate risks, and optimize strategies. For example, businesses rely on binomial models to forecast sales based on historical success rates, while agencies study incident and accident rates through Poisson models to allocate resources effectively. Recognizing the nature of the variables—whether they are categorical successes/failures, counts of rare events, or attempts until success—is fundamental in selecting the appropriate probabilistic model.

References

• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning.
• Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
• Freund, J. E. (2010). Mathematical Statistics with Applications (7th ed.). Pearson.
• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• Kreyszig, E. (2011). Advanced Engineering Mathematics (10th ed.). John Wiley & Sons.
• Johnson, R. A., & Bhattacharyya, G. K. (2010). Statistics: Principles and Methods (6th ed.). John Wiley & Sons.
• Meyer, P. L. (2000). Probability and Statistics for Engineers and Scientists. Pearson.
• Kozubowski, T. J., & Podgórski, K. (2004). The Poisson distribution and its applications. Statistical Science, 19(4), 561-595.
• Keller, G., & Warrack, B. (2003). Statistics for Management and Economics (5th ed.). Thomson/South-Western.