Probability Models

1. Probability Models Chapter 17

2. Bernoulli Trials (Binomial Setting) • 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 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. Check • Genetics says that children receive genes from each of their parents independently. Each child of a particular pair of parents has probability 0.25 of having type O blood. Suppose these parents have 5 children. Let X=the number of children with type O blood. Is this a binomial distribution?

5. Check • Shuffle a deck of cards. Turn over the first 10 cards, one at a time. Let Y=the number of aces you observe. Is this a binomial distribution?

6. Check • Shuffle a deck of cards. Turn over the top card. Put the card back in the deck, and shuffle again. Repeat this process until you get an ace. Let W=the number of cards required. Is this a binomial distribution?

7. The Binomial Model (Binomial Coefficient) • In n trials, there are ways to have k successes. • Read nCkas “n choose k.” • Note: n! = n(n – 1)  … 2  1, and n! is read as “n factorial.”

8. The Binomial Model 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) = nCxpxqn–x

9. 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…

10. The Normal Model to the Rescue! • 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

11. 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).

12. The Geometric Model Four Conditions for a Geometric Setting: • Binary: The possible outcomes of each trial can be classified as “success” or “failure”. • Independent: Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. • Trials: The goal is to count the number of trials until the first success occurs. • Success: On each trial, the probability p of success must be the same.

13. Situation • Your teacher is planning to give you 10 problems for homework. As an alternative, you can agree to play the birthday game. Here’s how it works. A student will be selected at random from your class and asked to guess the day of the week (for instance, Thurs) on which one of your teacher’s friends was born. If the student guesses correctly, then the class will have only one homework problem. If the student guesses the wrong day of the week, your teacher will once again select a student from the class at random. The chosen student will try to guess the day of the week on which a different one of your teacher’s friends was born. If this student gets it right, the class will have two homework problems. The game continues until a student correctly guesses the day on which one of your teacher’s (many) friends was born.

14. Trial • The random variable of interest in this game is Y=the number of guesses it takes to correctly match the birth day of one of your teacher’s friends. Each guess is one trial of the chance process. Let’s check the conditions for a geometric setting:

15. The Geometric Model 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

16. 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.

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. 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.

19. 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

20. What have we learned? • 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.