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Solution: D

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Chapter 17Probability Models

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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.

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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.

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The Geometric Model

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).

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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

E(X) 1

p2

q

p

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TI-Tips Geometric Probabilities

2nd DISTR Geometpdf(p,x) “pdf” stands for “probability density function” This allows us to find probabilities such as “The

probability of finding our 1st in the 5th box.”

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TI-Tips Geometric Probabilities

2nd DISTR Geometcdf(p,x) “cdf” stands for “cumulative density function” This allows us to find probabilities such as “The

probability of finding at least one by the 5th box.”

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Example: Postini is company specializing in communications security. The company monitors over 1 billion Internet messages per day and recently reported that 91% of e-mails are spam! Let’s assume that your e-mail is typical, 91% spam. We’ll also assume you aren’t using a spam filter, so every message gets dumped into your inbox. Since spam comes from many different sources, we’ll consider your messages to be independent. Overnight your inbox collects email. When you first check your e-mail in the morning, about how many spam e-mails should you expect to find before you find a real message? Find the probability that the 4th message in your inbox is the first one that isn’t spam.

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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).

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The Binomial Model (cont.)

In n trials, there are

ways to have k successes. Read nCk as “n choose k.”

Note: n! = n (n – 1) … 2 1, and n! is read as “n factorial.”

!

! !n k

nC

k n k

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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

np npq

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TI – Tips Binomial Probabilities

2nd DISTR Binompdf(n,p,X) This finds the probability of “finding exactly 2 in 5

boxes”

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TI – Tips Binomial Probabilities

2nd DISTR Binomcdf(n,p,X) This finds the probability of “finding up to 4 in 5

boxes” or “no more than 4 in 5 boxes” To find: “At least 5” = 1-binomcdf(n,p,4)

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Example: Postini reports that 91% of email messages are spam. Suppose your inbox contains 25 messages. What are the mean and standard deviation of the number of real messages you should expect to find in your inbox? What’s the probability that you’ll find only 1 or 2 real message?

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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…

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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

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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.

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Example: Postini reported that 91% of e-mail messages are spam. Recently, you installed a spam filter. You observe that over the past week it okayed only 151 of 1422 e-mails you received, classifying the resent as junk. Should you worry the filtering is too aggressive? What’s the probability that no more than 151 of 1422 e-mails is a real message?

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Sum it up!

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.

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Example: We want to know the probability that we find our first Tiger Woods picture in the fifth box of cereal. Tiger is in 20% of the boxes.

Example: Find the probability of getting a Tiger Woods picture by the time we open the fourth box of cereal.

Example: We want to know the probability of finding Tiger exactly twice among five boxes of cereal.

Example: Suppose we have ten boxes of cereal and we want to find the probability of finding up to 4 pictures of Tiger.

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Homework: Pg. 401 1-41 odd