Name ______

AP STATISTICS CHAPTER 8: THE BINOMIAL AND GEOMETRIC DISTRIBUTIONS

Ex: In a manufacturing plant, the probability that a randomly selected part is defective is 15%. In a random sample of 8 parts, what is the probability that exactly 2 are defective? What is the probability that 2 or less are defective?

A given setting is considered binomial if:

1. Each observation falls into one of just 2 categories:.

2. There are a of observations.

3. The observations .

4. The probability of success is .

If data are produced in a binomial setting, then the random variable X = number of success is called a

. We say that X is .

BINOMIAL PROBABILITY

If X has the binomial distribution with n observations and probability p of success on each observation, the probability of exactly k successes in the n observations is given by:

P ( X = k ) =

The number of inaccurate gauges in a group of four is a binomial random variable. If the probability of a defect is 0.1, what is the probability that only 1 is defective?

More than 1 is defective?

Graphing calculator commands:

A genetic trait of one family manifests itself in 25% of the offspring. If eight offspring are randomly selected, find the probability that the trait will appear in exactly three of them.

At least 5?

In a certain county, 30% of the voters are Republicans. If ten voters are selected at random, find the probability that no more than six of them will be Republicans.

What is the probability that at least 7 are not Republicans?

BINOMIAL PROBABILITY HOMEWORK

In a certain metropolitan area, nine out of ten households have a VCR. Let X denote the number among 8 randomly selected households that have a VCR.

Then X has the distribution.

1. What is the probability that exactly 5 of these homes will have a VCR?

2. What is the probability that exactly 7 of these homes will have a VCR?

3. What is the probability that 6 or more of these homes will have a VCR?

4. What is the probability that 5 or less of these homes will have a VCR?

Thirty percent of all automobiles undergoing an emission inspection at a certain inspection station fail the inspection.

1. What is the probability that exactly 5 of 10 randomly selected cars pass the inspection?

2. What is the probability that more than 8 of the next 10 cars will pass inspection?

BINOMIAL MEAN AND STANDARD DEVIATION

Given that X has the distribution, the mean () of X and standard deviation are given by:

Mean:

Standard deviation:

Ex: In a certain metropolitan area, nine out of ten households have a VCR. Let X denote the number among 8 randomly selected households that have a VCR.

Ex: Thirty percent of all automobiles undergoing an emission inspection at a certain inspection station fail the inspection. Fifty cars are inspected.

THE NORMAL APPROXIMATION

As the number of gets larger, the distribution gets close to a

distribution.

Rule of thumb: we can use the normal approximation when n and p satisfy:

1.

2.

Then, if X has the distribution , we can approximate it by using the distribution.

Ex: Some people believe that using cell phones will driving can be dangerous. A recent random survey asked 2000 U.S. adults if they agreed with the statement “Using a handheld cell phone while driving can reduce a driver’s reaction time.” Suppose that, in reality, 55% of all adult U.S. residents would say “agree” if asked this question. What is the probability that 1080 or less of the sample agree?

8.2 THE GEOMETRIC DISTRIBUTION

In a large fishtank, 30% of the goldfish are males. A customer desires a male goldfish, so the store employee scoops into the water and captures one fish. If the fish is not a male, he releases the fish into the tank and tries again. He continues scooping until he finds a male goldfish.

What is the probability that it takes 5 trials to get a male?

What is the probability distribution of X, the number of trials needed to find a male goldfish?

We say the random variable X has the geometric distribution if each observations falls into one of just two categories, or ; the probability of is the same for each observation; the observations are ; and the variable of interest if the number of required to obtain the first .

Given X has the geometric probability with probability p of success and ( 1- p ) of failure on each observation, the probability that the first success occurs on the nth trial is:

The mean of X is given by:

The variance of X is given by:

The probability that it takes more than n trials to see the first success is given by:

Ex: What is the probability that the pet store employee will need to scoop out 5 or more fish to find a male?

Graphing calculator commands:

What is the probability that the first son is the fourth child born?

What is the probability that the first son is born is at most four children?

A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person is 35%. What is the probability that the agent will sell the house to the third person she shows it to?

How many prospective buyers does she expect to show the house to before someone buys the house?

SUMMARY/QUESTIONS TO ASK IN CLASS