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 nCk as “n choose k.”
nCk can also be written 𝑛 𝑘
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 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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