### Binomial Probability Distributions

```Statistics Class 10
2/29/2012
Review
When playing roulette at the Bellagio casino in Las Vegas, a gambler is
trying to decide whether to bet \$5 on the number 13 or to bet \$5 that the
outcome is any one of these five possibilities: 0 or 00 or 1 or 2 or 3. From
Example 8, we know that the expected value of the \$5 bet for a single
number is -26₵. For the \$5 bet that the outcome is 0 or 00 or 1 or 2 or 3,
there is a probability of 5/38 of making a net profit of \$30 and a 33/38
probability of losing \$5.
a. Find the expected value for the \$5 bet that the outcome is 0 or 00 or 1
or 2 or 3.
b. Which bet is better: A \$5 bet on the number 13 or a \$5 bet that the
outcome is 0 or 00 or 1 or 2 or 3? Why?
Binomial Probability Distributions
A binomial probability distribution results from a procedure that meets all
the following requirements.
Binomial Probability Distributions
A binomial probability distribution results from a procedure that meets all
the following requirements.
1. The procedure has a fixed number of trials.
Binomial Probability Distributions
A binomial probability distribution results from a procedure that meets all
the following requirements.
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial
doesn’t affect the probabilities in the other trials.)
Binomial Probability Distributions
A binomial probability distribution results from a procedure that meets all
the following requirements.
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial
doesn’t affect the probabilities in the other trials.)
3. Each trial must have all outcomes classified into two categories
(commonly referred to as success and failure).
Binomial Probability Distributions
A binomial probability distribution results from a procedure that meets all
the following requirements.
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial
doesn’t affect the probabilities in the other trials.)
3. Each trial must have all outcomes classified into two categories
(commonly referred to as success and failure).
4. The probability of a success remains the same in all trials.
Binomial Probability Distributions
Note on Independence
Often when selecting a sample we do so without replacement. This means
that our events are dependent, and violate rule 2 of the binomial
probability distribution. However we can use the 5% guideline for
cumbersome calculations, and treat dependent events independent as long
as the sample size is no more than 5% of the population size.
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
• P(F)=q
q=probability of failure
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
• P(F)=q
q=probability of failure
• n
denotes the fixed number of trials
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
• P(F)=q
q=probability of failure
• n
denotes the fixed number of trials
• x
denotes a specific number of successes in n trials
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
• P(F)=q
q=probability of failure
• n
denotes the fixed number of trials
• x
denotes a specific number of successes in n trials
• p
denotes the probability of success in one of the n
trials
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
• P(F)=q
q=probability of failure
• n
denotes the fixed number of trials
• x
denotes a specific number of successes in n trials
• p
denotes the probability of success in one of the n
trials
• q
denotes the probability of failure in one of the n
trials
Binomial Probability Distributions
Notation for a Binomial Probability Distribution
S and F (success and failure) denote the two possible categories of
outcomes.
• P(S)=p
p=probability of success
• P(F)=q
q=probability of failure
• n
denotes the fixed number of trials
• x
denotes a specific number of successes in n trials
• p
denotes the probability of success in one of the n
trials
• q
denotes the probability of failure in one of the n
trials
• P(x)
denotes the probability of getting exactly x
successes among the n trials
Binomial Probability Distributions
Consider an experiment in which 5 offspring peas are
generated from 2 parents each having the green/yellow
combination of genes for pod color. The probability of an
offspring pea will have a green pod is ¾. That is P(green
pod) = 0.75. Suppose we want to find the probability that
exactly 3 of the 5 offspring peas have a green pod.
a. Does this procedure result in a binomial distribution?
b. If this procedure does result in a binomial distribution,
identify the values of , , ,  .
Binomial Probability Distributions
Consider an experiment in which 5 offspring peas are
generated from 2 parents each having the green/yellow
combination of genes for pod color. The probability of an
offspring pea will have a green pod is ¾. That is P(green
pod) = 0.75. Suppose we want to find the probability that
exactly 3 of the 5 offspring peas have a green pod.
a. Does this procedure result in a binomial distribution?
Yes
a. If this procedure does result in a binomial distribution,
identify the values of , , ,  .
n=5, x=3, p=0.75, and q=0.25
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Surveying 12 jurors and recording whether there
is a “no” response when they are asked if they have ever been
convicted of a felony
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Surveying 12 jurors and recording whether there
is a “no” response when they are asked if they have ever been
convicted of a felony
Binomial
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Treating 50 smokers with Nicorette and
recording whether there is a “yes” response when they are
asked if they experience any mouth or throat soreness.
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Treating 50 smokers with Nicorette and
recording whether there is a “yes” response when they are
asked if they experience any mouth or throat soreness.
Binomial
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Recording the number of children in 250 families
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Recording the number of children in 250 families
Not binomial, there are more than two outcomes.
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Fifteen different Governors are randomly
selected from the 50 Governors currently in office and the sex
of each Governor is recorded.
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Fifteen different Governors are randomly
selected from the 50 Governors currently in office and the sex
of each Governor is recorded.
Not binomial, not independent!
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Two hundred statistics students are randomly
selected and each is asked if he or she owns a Ti-84 Plus
Calculator.
Binomial Probability Distributions
Determine whether the given procedure results in a binomial
distribution. Two hundred statistics students are randomly
selected and each is asked if he or she owns a Ti-84 Plus
Calculator.
No, but yes?!?!? We can use the 5% guideline for
cumbersome calculations.
Binomial Probability Distributions
Binomial Probability Formula
In a binomial Probability distribution, probabilities can be calculated by
using the binomial probability formula.
Binomial Probability Distributions
Binomial Probability Formula
In a binomial Probability distribution, probabilities can be calculated by
using the binomial probability formula.
First recall/learn: Factorial symbol (!) denotes the product of decreasing
powers of positive whole numbers.
Binomial Probability Distributions
Binomial Probability Formula
In a binomial Probability distribution, probabilities can be calculated by
using the binomial probability formula.
First recall/learn: Factorial symbol (!) denotes the product of decreasing
powers of positive whole numbers. So 4! = 4 ∙ 3 ∙ 2 ∙ 1 and 0! = 1.
Binomial Probability Distributions
Binomial Probability Formula
In a binomial Probability distribution, probabilities can be calculated by
using the binomial probability formula.
First recall/learn: Factorial symbol (!) denotes the product of decreasing
powers of positive whole numbers. So 4! = 4 ∙ 3 ∙ 2 ∙ 1 and 0! = 1.
=
!
− !!
∙   ∙ − for  = 0,1,2, … ,
where
n=number of trials
x=number of success among n trials
(p=probability of success/q=probability of failure) in any one trial
Binomial Probability Distributions
The probability of an offspring pea will have a green pod is ¾.
That is P(green pod) = 0.75. Let’s use the binomial
probability formula to find probability that exactly 3 of the 5
offspring peas have a green pod.
Binomial Probability Distributions
The probability of an offspring pea will have a green pod is ¾.
That is P(green pod) = 0.75. Let’s use the binomial probability
formula to find probability that exactly 3 of the 5 offspring peas
have a green pod.
So n=5, x=3, p=0.75, and q=0.25
5!
3 =
⋅ 0.753 ⋅ 0.245−3 = 0.263671875
5 − 3 ! 3!
Binomial Probability Distributions
Assume that a procedure yields a binomial distribution with a trial
repeated 14 times. Use Table A-1 to find the probability of 4 successes
given the probability 0.60 of success on a single trial.
Binomial Probability Distributions
Assume that a procedure yields a binomial distribution with a trial
repeated 5 times. Using the Binomial Probability formula find the
probability of 2 successes given the probability .35 of success on a single
trial.
Binomial Probability Distributions
The brand name of McDonald’s has a 95% recognition rate. If a
McDonald’s executive wants to verify that rate by beginning with a small
sample of 15 randomly selected consumers, find the probability that
exactly 13 of the 15 consumers recognize the McDonald’s brand name.
Also find the probability that the number who recognize the brand name
is not 13.
Homework!!!
• 5-3: 1-8,13, 33, 35, and 39.
,  2 ,   for Binomial Distributions
For a Binomial Distribution ,  2 ,   are given by the following
formulas:
,  2 ,   for Binomial Distributions
For a Binomial Distribution ,  2 ,   are given by the following
formulas:
=∙
,  2 ,   for Binomial Distributions
For a Binomial Distribution ,  2 ,   are given by the following
formulas:
=∙
2 =  ∙  ∙
,  2 ,   for Binomial Distributions
For a Binomial Distribution ,  2 ,   are given by the following
formulas:
=∙
2 =  ∙  ∙
= ∙∙
,  2 ,   for Binomial Distributions
For a Binomial Distribution ,  2 ,   are given by the following
formulas:
=∙
2 =  ∙  ∙
= ∙∙
Now lets do example 1 and 2, then do problem 6 on the worksheet
,  2 ,   for Binomial Distributions
Use the given values of n and p to find the mean μ
and the standard deviation σ. Also, use the range rule
of thumb to find the minimum usual value μ - 2σ and
the maximum usual value μ + 2σ.
,  2 ,   for Binomial Distributions
Use the given values of n and p to find the mean μ
and the standard deviation σ. Also, use the range rule
of thumb to find the minimum usual value μ - 2σ and
the maximum usual value μ + 2σ.
Given: n = 60, p = 0.25
Mean: μ = np = (60)(.25) = 15
Standard deviation: σ = √(60 * .25 * .75) = 3.354
Min usual value: μ - 2σ = 15 – 2(3.354) = 8.292
Max usual value: μ + 2σ = 15 + 2(3.354) = 21.708
,  2 ,   for Binomial Distributions
Several economics students are unprepared for a
multiple-choice quiz with 25 questions, and all of
their answers are guesses. Each question has five
possible answers, and only one of them is correct.
Find the mean and standard deviation for the
number of correct answers for such students.
Mean: μ = (25)(1/5) = 5
Standard deviation: σ = √(25 *.2 *.8) = 2
,  2 ,   for Binomial Distributions
232 Mars, Inc., claims that 24% of its M&M plain
candies are blue. A sample of 100 M&Ms is randomly
selected. Find the mean and standard deviation for
the numbers of blue M&Ms in such groups of 100.
Mean: μ = np μ = (100)(.24) = 24
Standard deviation: σ = √(100 *.24 *.76) = 4.3
,  2 ,   for Binomial Distributions
Data Set 18 in Appendix B consists of a random
sample of 100 M&Ms in which 27 are blue. Is this
result unusual? Does it seem that the claimed rate of
24% is wrong?
μ = 24 σ = 4.3
The max usual values: 24 + 2(4.3) = 32.6 M&Ms
The min usual values: 24 – 2(4.3) = 15.4 M&Ms
Homework!!!
• 5-3: 1-8,13, 33, 35, and 39.
• 5-4: 1-12, 17, 19
```