Chapter 17 Probability Models What I will know and be able to do Know the appropriate conditions under which a Geometric, Binomial, or Normal model is used and calculate Geometric and Binomial probabilities. Assignment: Read Chapter 17

Chapter 17

Probability Models

What I will know and be able to do

Know the appropriate conditions under which a

Geometric, Binomial, or Normal model is used and

calculate Geometric and Binomial probabilities.

Assignment:

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.

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

Teaching Tip

■ The term Bernoulli is not officially an AP topic. However, it is a good

way to begin these topics. Don’t worry if students forget this name as you focus on the multiplication principle and binomial

Example 1: Bernoulli or not…..

■ 1. You are rolling 5 die and need to get at least two 6’s to win the

game.

■ 2. We record the distribution of eye colors found in a group of 500

people.

■ 3. A city council of 11 Republicans and 8 Democrats picks a

committee of 4 at random. What’s the probability they choose all Democrats?

■ 4. A 2002 Rutgers University study found that 74% of high school

students have cheated on a test at least once. Mr. McDonough

Example 2: Simulations

20% of cereal boxes contain pictures of LeBron James, 30% a picture of Danica Patrick and the rest a picture of Serena Williams. You are opening boxes of cereal one at a time looking for LeBron James’

picture. You want to know how many boxes you might have to open in order to find LeBron.

a. Describe how you would simulate the search using a random digit table.

b. Run 30 trials.

c. Based on the simulation, estimate the probabilities that you

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

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

Teaching tip

■ Geometric probabilities are simply an extension of the

multiplication principle. Do not focus on geometric formula, but rather on what students already know from the

previous chapter.

■ In fact, students already performed geometric probability

calculations in Chapter 13 (#39 and following). You might want to start class with students redoing a couple of those problems, identifying them as geometric, and then diving into the binomial model as something they don’t know how to solve yet.

■ For example, transition from “What are the chances he

Example 3: Speckled Skittles

A new gimmick has 30% of Skittles candies covered with

speckles. These “groovy” candies are mixed randomly with the normal candies as they are put into the bags for distribution and sale. You buy a bag and remove candies one at a time looking for speckles.

a. What is the probability that the first speckled one we see is the

fourth candy we get?

b. What’s the probability that the first speckled one is the tenth

one?

c. Write the general formula.

d. How many do we expect to have to check, on average, to find

a speckled one?

Slide 17- 11

Example 3 answers

☺

a. P(first speckled is fourth candy)=(.7)3(.3)=.1029 b. P(first speckled is tenth candy)=(.7)10(.3)=.0085 c. General Formula: P(X=x)=qx-1p

Slide 17- 12

Example 4: Jumper Cables

Suppose that 40% of the students who drive to Wilson HS carry

jumper cables in their car. Your car has a dead battery and you don’t have jumper cables, so you decide to stop students who are headed to the parking lot and ask them whether they have a pair.

a. What is the probability that the third student you ask will have

jumper cables?

b. What is the probability that first student you ask will have jumper

cables?

c. What is the probability that three or fewer students must be

stopped to find cables?

d. What is the average number of students you would have to stop to

Slide 17- 13

Example 4 answers

☺

a. P(first person with cables is third one asked)=(.6)2(.4)=.144 b. P(first person with cables is first one asked)=(.6)0(.4)=.4

c. P(x≤3)=P(1 or 2 or 3)=P(1) +P(2)+P(3) = (.6)0(.4)+ (.6)1(.4)+ (.6)2(.4)

=.4 + .24 + .144 = .784

Slide 17- 14

Geometric Model with Calculator

geometpdf(p,x)

p is the probability of success

x is the number you are “waiting” for

Based on the last example, what is the probability of the 4th student have the jumper cables you need.

geometpdf(.4,4) = .0864

What is the probability that you will find jumper cables in 4 or fewer students asked?

geometcdf(.4,4) = .8704 …….or………

Slide 17- 15

Example: Skittles again!

Back to the speckled skittles....what is the probability that

we’ll find two speckled ones in a handful of five candies?

This is a little bit different…notice we are not waiting until

something “special” happens…

Slide 17- 16

So…How do you find the answer?

■ Would this work?

P(2 speckled out of 5) = (.3)2(.7)3

NO

Why not? This method says that the first 2 would be speckled and the last three are not. Can you get 2 speckled in another way? We need to count the number of ways that getting 2 speckled out of 5 can happen.

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

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

The Binomial Model (cont.)

■

In

n

trials, there are

ways to have

k

successes.

■

Read

n

C

as “

n

choose

k.

”

■

Note: , and

n

! is read as “

n

Slide 17- 20

Back to the Skittles….

Now lets solve it:

P(2 speckled out of 5) =

C

(.3)

(.7)

Slide 17- 21

Calculator for Binomial Models

binompdf(n,p,x)

n is the number in the group p is the probability of success

x is the number of successes you want

Based on the last example, what is the probability of 2 out 5 being speckled? P(X=2) = binompdf(5,.3,2) =

What is the probability that you will get 2 or fewer speckled Skittles out of 5? P(X<=2) = binomcdf(5,.3,2) =

Slide 17- 22

Before you do a problem:

■ Check to see if there are Bernoulli trials.

1. 2. 3.

■ If you can’t say “independent”, you must

show

to be true.

■ State the model that you are using, with correct symbolic

notation.(Geom(p) or Binom(n,p))

Example: Blood Donors

People with O-negative blood are called “universal” donors because O-negative blood can be given to anyone else, regardless of the recipient’s blood type. Only about 6% of people have O-negative blood.

a. 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?

b. What’s the probability that the first O-negative donor found is the one of the first four people in line?

c. What is the probability that there are 4 O-negative donors amount in a group of 50 donors standing in line?

d. What is the probability that 7 out of the 100 donors that donated blood last Wednesday were O-negative?

e. What is the probability that 10 or fewer donors out of a 87 donor pool are O-negative? f. What is the probability that 4 or 5 of the donors out of a group 63 donors are O-negative? g. What is the probability that at least 1 out of 20 donors is type O-negative?

Slide 17- 24

Blood Example

What do you have to do first? Lots of stuff…

■ E(X)=16.7

■ P(X<=4)=.2193

Teaching Tips

■ The binomial formulas are on the official AP formula sheet. Which is

provided for the entire AP exam. Give your students a copy!

■ Note: The normal approximation of the binomial distribution is an

optional AP topic. It is covered (differently) through sampling

Example – More Blood

Suppose the Red Cross anticipates the need for at least 1850 units of O-negative blood this year. It estimates that it will collect blood from 32,000 donors. How great is the risk that they will fall short of

meeting its need?

We need to calculate the probability of getting exactly 1850 units of O-neg blood from 32,000 donors. What would that be?

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).

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:

Slide 17- 29

Blood and more blood….

Check Bernoulli and success failure condition:

np ≥ 10 and nq ≥ 10

32000(.06) ≥ 10 and 32000(.94) ≥ 10 ≥ 10 ≥ 10

So, we can use a Normal model, using the mean and standard deviation from the Binomial model.

μ = np = 32000(.06) = 1920 and SD=

N(1920,42.48)

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

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

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

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

AP Tips

■ The AP rubrics usually have three requirements for probability

problems:

■ Name of distribution

AP Tips

■ The Name can be identified with ■ words (binomial) or with

■ the proper formula (P(X = x) =

nCx p

x _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

Assn #4L Even Answers

#8.) a.) 0.0184

#10.) 50 chips

b.) 0.183

#12.) a.) 12.5 people

b.) 0.716

c.) 0.0527

d.) 0.528

#20.) a.) 0.201

b.) 0.453

c.) 0.171

d.) 0.989

#26.) Binomial model:

P(more than 265) = .116

Normal model:

Assn #4L Even Answers –Problem 26

These may be considered Bernoulli Trials –

■

There are only two possible outcomes: showing up or not

showing up.

■

The probability is constant (about 95%)

■

The trials are not independent, but a sample of 275 is a lot

smaller than 10% of all passengers.

■

Therefore it can be modeled by Binom(275, 0.95)

■

According to the Binomial model the

P(more than 265) = 0.116