The Binomial Distribution

1. What are the conditions for the Binomial Setting?

2. What is the Binomial Distribution Notation?

3. How do you compute the probability of a value of X using the Binomial Distribution?

4. How do you find the means and standard deviation of the Binomial Distribution?

The Binomial DistributionProperties of a Binomial Experiment

1. It consists of a fixed number of observations called trials.2. Each trial can result in one of only two mutually exclusive outcomes labeled success (S) and failure (F).3. Outcomes of different trials are independent.4. The probability that a trial results in S is the same for each trial.

The binomial random variable X is defined as X = number of successes observed when

experiment is performed

The probability distribution of X is called the binomial probability distribution.

Binomial Distribution

1. There are only two possible outcomes for each trial. (success or failure)

Binomial Distribution

2. Each trial is independent.

Binomial Distribution

3. The probability for success for each trial is the same for each trial or observation.

=P( ) P( )

Binomial Distribution

4. There are a set number of trials.

Determining Binomial Conditions

• Work on the worksheet• Answers:

!

!( )!

n

n kk

nC

k n k

Binomial Distribution Notation

X is B(n,p)n is the number of observations

p is the probability of success of any one observation

Example

• Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X off successes in 5 independent observations.

• How would you describe this with “B” notation?• X=B(5,.25)

* n = total # of trials* x or k = # of success* p = probability of success* 1-p = probability of failure* n-k = # of failures

0.2270)4(.)6(.!5!7

!12)7(p 57

ExampleThe adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that precisely 7 jurors are black.

Clearly, n=12 and p =.6, so

792119854321

12111098

)54321)(7654321(

121110987654321

!5!7

!12:Note

Example - continued

The adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that less than 3 are black.

Clearly, n = 12 and p= 0.6, so

0 12 1 11 2 10

P(x 3) P(x 2) p(0) p(1) p(2)

12! 12! 12!(.6) (.4) (.6) (.4) (.6) (.4)

0!12! 1!11! 2!10!0.00002 0.00031 0.00249 0.00281

Using the Calculator• To find the binomial probability using

the calculator, use the “binompdf” function.

• The suffix “pdf” stands for probability distribution function.

• The binompdf command is found under 2nd/DISTR/0:binompdf

• Type in the calculator binompdf(n, p, X).

Example – w/Calculator

• The adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that precisely 7 jurors are black.

• Binompdf(12, .6, 7) =

Example – w/Calculator

• The adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that less than 3 are black.

• Binompdf(12, .6, 2) =

• +Binompdf(12, .6, 1) =

• +Binompdf(12, .6, 0) =

Calculator Practice

Calculate the following probabilities using the Binomialpdf on the calculator.1.n=6, p=.03, k=32.n=4, p=.18, k=23.n=5, p=.63, k=34.n=9, p=.42, k=05.n=10, p=.37, k=5

Cumulative Binomial Probability

• The cumulative distribution function (cdf) of X calculates the sum of the probabilities of 0, 1, 2, … up to a value of X.

• In other words, it calculates the probability of obtaining at most X successes in n trials. (Less than or equal to)

• Calculator command binomcdf(n, p, k) = P(X≤ k).

Example Revisited

• The adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that less than 3 are black.

• P(X < 3)…. Or, in terms of cdf: P(X ≤ 2)

• Binomcdf(12, .6, 2) =

More Examples• Historically, 32% of Mr. Fribush’s AP Stat

students score at least a 4 on the AP Exam. Assuming each student has an equally likely chance of scoring at least a 4 and each student is independent of one another, find the following probabilities for a class of 25 students:

• The probability that exactly 10 students score a 4 or better.

• The probability that less than 10 students score a 4 or better.

• The probability that at most 10 students score a 4 or better.

• The probability that more than 10 students score a 4 or better.

• The probability that at least 10 students score a 4 or better.

Homework

• Worksheet