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Chapter 17

Chapter 17. Probability Models. Bernoulli Trials. The basis for the probability models we will examine in this chapter is the Bernoulli trial . We have Bernoulli trials if: there are two possible outcomes success and failure the probability of success, p , is constant .

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Chapter 17

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  1. Chapter 17 Probability Models

  2. Bernoulli Trials • The basis for the probability models we will examine in this chapter is the Bernoulli trial. • We have Bernoulli trials if: • there are two possible outcomes • success and failure • the probability of success, p, is constant. • the trials are independent.

  3. The Geometric Model • A single Bernoulli trial is usually not all that interesting. • A Geometric probability model tells us the probability for a random variable that counts the number of Bernoulli trials until the first success. • Geometric models are completely specified by one parameter, p, the probability of success, and are denoted Geom(p).

  4. The Geometric Model (cont.) Geometric probability model for Bernoulli trials: Geom(p) p = probability of success q = 1 – p = probability of failure X = number of trials until the first success occurs P(X = x) = qx-1p

  5. Geometric Model - Example • Suppose you are shooting free throws. After rigorous data collection and calculations, you find that your probability of making a free throw to be 0.3. • Assume that you meet the conditions for Bernoulli trials. • What is the probability you will make your first basket on the 4th shot? • What is the probability you will make your first basket before or on the 4th shot?

  6. Let’s take a POP quiz! Suppose you come to class to find you are having a pop quiz. The quiz has only four multiple choice questions. You have not had time to prepare for this quiz so you are completely guessing for each question. There are a total of 5 choices (one of which is right) for each question. What is the probability that you get exactly three correct?

  7. The Binomial Model • A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of trials. • Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p).

  8. Is this a situation for the Binomial? • Determining whether each of 3000 heart pacemakers is acceptable or defective. • Surveying people and asking them what they think of the current president. • Spinning the roulette wheel 12 times and finding the number of times that the outcome is an odd number.

  9. The Binomial Model • In n trials, there are ways to have x successes. • Read nCx as “n choose x,” and is called a combination. • Note: n! = n x(n – 1)x … x 2 x 1, and n! is read as “n factorial.”

  10. The Binomial Model (cont.) n = number of trials p = probability of success q = 1 – p = probability of failure x = number of successes in n trials

  11. Example A report from the Secretary of Health and Human Services stated that 70% of single-vehicle traffic fatalities that occur at night on weekends involve an intoxicated driver. If a sample of 10 single-vehicle traffic fatalities that occur at night on a weekend is selected, find the probability that exactly 5 involve a driver that is intoxicated.

  12. Using table generated in MINITAB P(x=5) = 0.102919

  13. How about the probability of at least 8 involving an intoxicated driver?

  14. You try this one! The participants in a television quiz show are picked from a large pool of applicants with approximately equal numbers of men and women. Among the last 10 participants there have been only 2 women. If participants are picked randomly, what is the probability of getting 2 or fewer women when 10 people are picked?

  15. The Normal Model to the Rescue! • When dealing with a large number of trials in a Binomial situation, making direct calculations of the probabilities becomes tedious (or outright impossible). • Fortunately, the Normal model comes to the rescue…

  16. The Normal Model to the Rescue (cont.) • As long as the Success/Failure Condition holds, we can use the Normal model to approximate Binomial probabilities. • Success/failure condition: A Binomial model is approximately Normal if we expect at least 10 successes and 10 failures: np ≥ 10 and nq ≥ 10.

  17. Continuous Random Variables • When we use the Normal model to approximate the Binomial model, we are using a continuous random variable to approximate a discrete random variable. • So, when we use the Normal model, we no longer calculate the probability that the random variable equals a particular value, but only that it lies between two values.

  18. Example: An Olympic archer is able to hit the bull’s-eye 80% of the time. Assume that each shot it independent of the others. She will be shooting 200 arrows in a large competition. a. What are the mean and standard deviation for the number of bull’s-eyes she might get? b. Is the normal model appropriate here? c. Use the 68-95-99.7% Rule to describe the distribution of the number of bull’s-eye she might get. d. Would you be surprised if she only made 140 bull’s-eyes? Explain.

  19. The Poisson Model • The Poisson probability model was originally derived to approximate the Binomial model when the probability of success, p, is very small and the number of trials, n, is very large. • The parameter for the Poisson model is λ. To approximate a Binomial model with a Poisson model, just make their means match: λ = np.

  20. The Poisson Model (cont.) Poisson probability model for successes: Poisson(λ) λ = mean number of successes X = number of successes e is an important mathematical constant (approximately 2.71828)

  21. The Poisson Model (cont.) • Although it was originally an approximation to the Binomial, the Poisson model is also used directly to model the probability of the occurrence of events for a variety of phenomena. • It’s a good model to consider whenever your data consist of counts of occurrences. • It requires only that the events be independent and that the mean number of occurrences stays constant.

  22. What Can Go Wrong? • Be sure you have Bernoulli trials. • You need two outcomes per trial, a constant probability of success, and independence. • Remember that the 10% Condition provides a reasonable substitute for independence. • Don’t confuse Geometric and Binomial models. • Don’t use the Normal approximation with small n. • You need at least 10 successes and 10 failures to use the Normal approximation.

  23. What have we learned? • Bernoulli trials show up in lots of places. • Depending on the random variable of interest, we might be dealing with a • Geometric model • Binomial model • Normal model • Poisson model

  24. What have we learned? (cont.) • Geometric model • When we’re interested in the number of Bernoulli trials until the next success. • Binomial model • When we’re interested in the number of successes in a certain number of Bernoulli trials. • Normal model • To approximate a Binomial model when we expect at least 10 successes and 10 failures. • Poisson model • To approximate a Binomial model when the probability of success, p, is very small and the number of trials, n, is very large.

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