Binomial vs. Geometric

1
Binomial vs. Geometric

• Chapter 8
• Binomial and Geometric Distributions

2
Binomial vs. Geometric
The Binomial Setting The Geometric Setting

• Each observation falls into
• one of two categories.

1. Each observation falls into one of two
categories.

• The probability of success
• is the same for each
• observation.

2. The probability of success is the same
for each observation.

• The observations are all
• independent.

• The observations are all
• independent.

• There is a fixed number n
• of observations.

4. The variable of interest is the number of
trials required to obtain the 1st success.
3
Are Random Variables and Binomial Distributions
Linked?
RECALL Suppose that each of three randomly
selected customers purchasing a hot tub at a
certain store chooses either an electric (E) or a
gas (G) model. Assume that these customers make
their choices independently of one another and
that 40 of all customers select an electric
model. The number among the three customers who
purchase an electric hot tub is a random
variable. What is the probability
distribution?
4
Are Random Variables and Binomial Distributions
Linked?
X number of people who purchase electric hot tub
X
0 1 2 3
.288
P(X)
.216
.432
.064
(.6)(.6)(.6)
GGG
EEG GEE EGE
(.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4)
EGG GEG GGE
(.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4)
EEE
(.4)(.4)(.4)
5
Combinations
Formula
Practice
6
Developing the Formula
Ex. 8.1 Blood Type pg. 440 Pract. Of Stats
2nd Ed Blood type is inherited. If both parents
carry genes for the O and A blood types, each
child has a probability 0.25 of getting two O
genes, thus having blood type O. Let X the
number of O blood types among 3 children.
Different children inherit genes independently of
each other. Since X has the binomial distribution
with n 3 and p 0.25. We say that X is B(3,
0.25)
7
Developing the Formula
Outcomes
Probability
Rewritten
OcOcOc
OOcOc
OcOOc
OcOcO
OOOc
OOcO
OcOO
OOO
8
Developing the Formula
Be able to recognize the formula even though you
can set up a Prob. Distribution without it.
Rewritten
n of observations p probablity of success k
given value of variable
9
Working with probability distributions

• State the distribution to be used
• State important numbers
• Binomial n p
• Geometric p
• Define the variable
• Write proper probability notation

10
Twenty-five percent of the customers entering a
grocery store between 5 p.m. and 7 p.m. use an
express checkout. Consider five randomly
selected customers, and let X denote the number
among the five who use the express checkout.
binomial
n 5 p .25
X of people use express
11
What is the probability that two used the express
checkout?
Get started with the required components

• Binomial, n 5, p .25
• X of people use express
• P(X 2)

12
Calculator option 1nCr n choose r 5 math
prb 3nCr 2 enter
REQUIRED COMPONENTS 1, 2, 3

• Define the situation Binomial, with n 5 and p
  0.25
• Identify X let X of people use express
• Write proper probability notation that the
• question is asking P( X 2)

Calculator option 2 2nd vars 0binompdf(5,
0.25, 2) .2637
13
What is the probability that no more than three
used express checkout?

• Binomial, n 5, p .25
• X of people use express
• P(X 3)

P(X 3)P(X0)P(X1)P(X2)P(X3)
Calculator option P(X 3) binomcdf(5, 0.25,
3) .9844
14
What is the probability that at least four used
express checkout?

• Binomial, n 5, p .25
• X of people use express
• P(X 4)

Calculator option P(X 4) 1-P(Xlt4)
1-binomcdf(5, 0.25, 3) .0156
15
Do you believe your children will have a higher
standard of living than you have? This question
was asked to a national sample of American adults
with children in a Time/CNN poll (1/29,96).
Assume that the true percentage of all American
adults who believe their children will have a
higher standard of living is .60. Let X represent
the number who believe their children will have a
higher standard of livingfrom a random sample of
8 American adults.
binomial
n 8 p .60
X of people who believe
16
Interpret P(X 3) and find the numerical answer.
binomial
n 8 p .60
X of people who believe
The probability that 3 of the people from the
random sample of 8 believe their children
will have a higher standard of living.
17
Interpret P(X 3) and find the numerical answer.
binomial
n 8 p .60
X of people who believe
P(X3) binomialpdf(8, .6, 3) .1239 or
18
Find the probability that none of the parents
believe their children will have a higher
standard.

• binomial, n 8, p 0.60

• X of people who believe
• P(X 0 )

19
Find the probability that none of the parents
believe their children will have a higher
standard.
binomial
n 8 p .60
X of people who believe
or P(X0)
binomialpdf(8, .6, 0) .00066
20
Binomial vs. Geometric
The Binomial Setting The Geometric Setting

• Each observation falls into
• one of two categories.

1. Each observation falls into one of two
categories.

• The probability of success
• is the same for each
• observation.

2. The probability of success is the same
for each observation.

• The observations are all
• independent.

• The observations are all
• independent.

• There is a fixed number n
• of observations.

4. The variable of interest is the number of
trials required to obtain the 1st success.
21
Developing the Geometric Formula
Probability
X
22
The Mean and Standard Deviation of a Geometric
Random Variable

• If X is a geometric random variable with
  probability of success p on each trial, the
  expected value of the random variable (the
  expected number of trials to get the first
  success) is

X
23
Suppose we have data that suggest that 3 of a
companys hard disc drives are defective. You
have been asked to determine the probability that
the first defective hard drive is the fifth unit
tested.

• geometric

p .03

• X of disc drives till defective
• Find P(X 5)

24
A basketball player makes 80 of her free throws.
We put her on the free throw line and ask her to
shoot free throws until she misses one. Let X
the number of free throws the player takes until
she misses.

• Geometric,

p .20

• X of free throws till miss

25
What is the probability that she will make 5
shots before she misses? (When is the 1st
success?)
geometric
p .20
X of free throws till miss
What is the probability that she will miss 5
shots before she makes one?
geometric
p .80
Y of free throws till make
26
What is the probability that she will make at
most 5 shots before she misses?
geometric
p .20
X of free throws till miss
Or 1 P(makes all 6 shots) 1 (.806) .7379