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. the trials are independent.

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).

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

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?

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?

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).

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.

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.”

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

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.

Using table generated in MINITAB x P(x) 0.000005 1 0.000138 2 0.001447 3 0.009002 4 0.036757 5 0.102919 6 0.200121 7 0.266828 8 0.233474 9 0.121061 10 0.028248 Using table generated in MINITAB P(x=5) = 0.102919

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

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?

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…

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.

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.

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.

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.

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)

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.

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.

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

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.