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?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 8: THE BINOMIAL AND GEOMETRIC DISTRIBUTIONS
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?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 8: THE BINOMIAL AND GEOMETRIC DISTRIBUTIONS
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?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 8: THE BINOMIAL AND GEOMETRIC DISTRIBUTIONS
BINOMIAL MEAN AND STANDARD
DEVIATION
Given that X has the
distribution, the mean (
and standard deviation are given by:
) of X
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?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 8: THE BINOMIAL AND GEOMETRIC DISTRIBUTIONS
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?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 8: THE BINOMIAL AND GEOMETRIC DISTRIBUTIONS
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