Discrete probability functions(Chapter 17)

There are two useful probability functions that have a lot of applications…

But first! Let’s cover some prerequisites: If you line up five people in order, how many

different ways can you do that? If you pick two people out of a class of 28 for

president and vice-president, how many ways to do this are there?

If you simply pick two people to be student representatives from a class of 28, there will only be 378 possibilities. Why?

(28)(27) = 756

5! = 120

Combinations

The number of ways you can pick k people out of n without regard to order is

With our example: On your calculator it is nCr under the

MATH:PROB menu. Try this: 28 nCr 2

!

!( )!n k

n nC

k k n k

28! 28! 28 27

2!(28 2)! 2!26! 2

Bernoulli trials

First, we say that an event is a Bernoulli trial if There are two possible outcomes (say Heads

or Tails) The probability of success is constant (does

not change) The trials are independent of each other

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 because you are picking without replacement. 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.

Are these “Bernoulli trials?”

You are rolling five dice and need to get at least two sixes to win the game.

We record the eye colors found in a group of 500 people.

A toy company recalls a doll because about 3% explode when touched. Customers return 37 of these dolls. Is the company likely to find any exploding ones?

A city council picks a committee of 4 at random from 20 candidates, 10 men and 10 women. What’s the probability that they are all men?

This is a Bernoulli trial

This is NOT a Bernoulli trial – more than two outcomes

This is a Bernoulli trial

This is NOT a Bernoulli – the probability changes.

The binomial function

The BINOMIAL probability model is appropriate for a random variable that counts the number of successes in a fixed number of Bernoulli trials.

Example: What is the probability you will roll exactly four 6’s in ten die rolls?

The binomial probability function looks like this:

where k is the number of successes and p is the probability of success.

On the calculator, we’ll use binompdf(n,p,k)

(1 )k n knp p

k

The geometric function

The GEOMETRIC probability model is appropriate for a random variable that is a count of the number of Bernoulli trials until the first success.

Example: What is probability you will roll a 6 for the first time on the fourth die roll?

The geometric probability function looks like this:

where k is the number of trials until the first success, and p is the probability of success.

On the calculator, try geompdf(p,k) but it’s often easier to use the homescreen.

1(1 )kp p

Example 1

There are 16 people in the room. The probability of finding left-handers is 13%.

What is probability that exactly three are` left handed?

What is the probability that at most 7 of the 16 are left-handed?

What is probability that the first lefthander you pick will be on the fourth trial?

Example 2

Suppose that 75% of all drivers always wear seatbelts.

Of ten drivers [in their cars] waiting at a signal, what is probability that more than half of the drivers are NOT wearing a seatbelt?

What is probability that the first unbelted driver is in the 3rd car we observe?

Example 3

On a multiple choice test with five choices per question, there are 40 problems.

What is the probability that a person guessing randomly will answer all of them incorrectly?

What is probability that a person guessing randomly will get AT MOST half of them correct?

The mean and standard deviationIn the Binomial probability

distribution, the mean and standard deviation are given by

In the Geometric probability distribution, we’ll only need the mean…

But here’s the standard deviation anyway.

1

B

B

np

np p npq

2

1G

p

p

1G p

The mean and standard deviation in practice…As you all know, about

8% of all males are colorblind. Suppose a researcher is looking for colorblind subjects for a secret experiment.

How many men should the researcher expect to check before he finds the first colorblind man?

1

B

B

np

np p npq

1G p

112.5

.08G

The mean and standard deviation in practice…As you all know, about 8% of

all males are colorblind. Suppose a researcher is looking for colorblind subjects for a secret experiment.

How many colorblind men should there be among 200 subjects?

What about the standard deviation?

1

B

B

np

np p npq

(200)(.08) 16

(200)(.08)(.92) 3.84

B

B

The Normal approximation to the Binomial Sometimes, the numbers in a Binomial distribution

are too large for computation. (For example, try 75! on your calculator.)

When the numbers are too large, we can use the Normal distribution in place of the Binomial.

But there are conditions you have to meet first…

If a variable has a Binomial distribution, and if np and n(1-p) are both greater than or equal to 10,

then we can replaceBinomcdf(n,p,k)

with Normalcdf(-∞,k,np,sqrt(npq))

The apple orchard problem An orchard owner knows that he’ll have to use

about 6% of his harvest for cider because they will have bruises or blemishes. He expects one tree to produce about 300 apples.

a) What probability function is appropriate here?

b) What is the mean and standard deviation?

c) What is the probability that the tree has no more than 12 cider apples?

d) Is it likely that there will be more than 50 cider apples?