Probability Models for Bernoulli Trials: Geometric, Binomial, and Normal

1. Chapter 16 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 (fixed). • 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. 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.

6. 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).

7. The Binomial Model (cont.) • In n trials, there are ways to have k successes. • Read nCkas “n choose k.” • nCk can also be written • Note: , and n! is read as “n factorial.”

8. 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) = nCx px qn–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 (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 See example on pages 425-426 in your text book.

11. 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.

12. 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.

13. 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

14. 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.

15. AP Tips • The AP rubrics usually have three requirements for probability problems: • Name of distribution • Identification of correct parameters • Correct calculation

16. AP Tips • The Name can be identified with • words (binomial) or with • the proper formula (P(X = x) = nCx px qn–x) • (Your teacher will probably ask you to write both and that’s a good thing!) • The parameters should use standard notation: • n = 10, p = 0.7 or • μ = 65”, σ = 3” (for a normal problem)

17. Examples A new sales gimmick has 30% of M&M’s covered with speckles. These new candies are mixed randomly with normal M&M’s as they are put in bags for distribution and sale. You buy a bag a remove candies one at a time looking for speckles. • What’s the probability that the first speckled M&M we see is the fourth on removed from the bag? The 10th? • What’s the probability that the first speckled M&M is one of the first three? • How many M&M’s do we expect to have to check to find a speckled one? • What’s the probability that we will find 2 speckled M&M’s in a handful of 5?

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