Chapter 17 Probability Models Overview Key Concepts Know how to tell if a situation involves Bernoulli trials. Be able to choose whether to use a Geometric or a Binomial model for a random variable involving Bernoulli trials. Know the appropriate conditions for using a Geometric, Binomial, or Normal model. Know how to find the expected value of a Geometric model. Be able to calculate Binomial probabilities, perhaps estimating with a Normal model. Be able to interpret means, standard deviations, and probabilities in the Bernoulli trial context. 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. 1
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 = # of trials until the first success occurs P(X = x) = q x-1 p 1 p q 2 p Independence One of the important requirements for Bernoulli trials is that the trials be independent. When we don t have an infinite population, the trials are not independent. But, there is a rule that allows us to pretend we have independent trials: The 10% condition: Bernoulli trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population. 2
Requirements for Geometric Models There are four requirements for geometric models. They are: 1. There are only two possible outcomes (success or failure). 2. The probability of success is the same on every trial (denoted p) 3. The trials are independent. 4. The variable of interest is the number of trials it takes to reach a success Example People with O-negative blood are called the universal donors because O-negative blood can be given to anyone else. Only about 6% of people have O-negative blood. If donors line up at random for a blood drive, how many do you expect to examine before you find someone who has O-negative blood? What s the probability that the first O- negative donor is the fourth person? Same Problem With the Calculator The calculator will calculate the probability of a success on a given trial. 2 nd VARS geometpdf(p,x) The calculator will also calculate the cumulative probability 2 nd VARS geometcdf(p,x) This calculates the probability of finding the first success on or before the x trial. 3
Homework Assignment P. 383 33, 37 & P. 398 3, 5, 7 Homework Check Tomorrow New Situation Still using the cereal setup, lets change what we are looking for. If we buy exactly 5 boxes of cereal, what s the probability you get exactly 2 pictures of Tiger Woods? In other words, now we are interested in the number of successes in 5 trials. 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 Bernoulli trials. Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p). 4
Binomial Probability Models With Binomial models, a success is a success no matter what trial it occurs on. So saying 5 trials with 2 successes means that the two successes can come on any of the trials. We need to determine how many different combinations of 2 out of 5 successes there are. The Binomial Model (cont.) In n trials, there are n! Ck k! n k! ways to have k successes ( n C k as n choose k. ) n Note: n! = n x (n-1) x x 2 x 1, and n! is read as n factorial. Use your calculator to answer 5 choose 2. (Math PRB 3) The Binomial Model (cont.) Binomial probability model for Bernoulli trials: Binom(n,p) n = number of trials p = probability of success q = 1 p = probability of failure X = # of successes in n trials P(X = x) = n C x p x q n-x np npq 5
Requirements for Binomial Models There are four requirements for binomial models. They are: 1. There are only two possible outcomes (success or failure). 2. The probability of success is the same on every trial (denoted p). 3. The trials are independent. 4. There is a fixed number of trials (n). Example Suppose 20 donors come to the blood drive. What are the mean and standard deviation of the number of universal donors among them? What is the probability that there are 2 or 3 universal donors? Calculator Steps The calculator will calculate the probability of a number of successes in n trials. 2 nd DISTR binompdf(n,p,x) n = total number of trials p = probability of success x = number of successes The calculator will also calculate the cumulative probability. 2 nd DISTR bionomcdf(n,p,x) This calculates the probability of getting x or fewer successes among the n trials. 6
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. 7
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? (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. Homework Assignment P. 398 13, 15, 17, 21, 25 & P. 399 19, 25, 31 8