Probability Probability DistributionsDistributions

BINOMIAL BINOMIAL DISTRIBUTIONDISTRIBUTION

Binomial TrialsBinomial Trials There are a specified number of There are a specified number of

repeatedrepeated,, independentindependent trialstrials The outcome of Binomial experiments or The outcome of Binomial experiments or

Trials are either a Trials are either a SuccessSuccess or or FailureFailure The Probability of Success is the The Probability of Success is the samesame in in

each trialeach trial The The Random VariableRandom Variable is the number of is the number of

successes in a given number of trialssuccesses in a given number of trials

These are also called Bernoulli TrialsThese are also called Bernoulli Trials

ExamplesExamplesA coin is flippedA coin is flipped

A test of true and false questionsA test of true and false questions

A test of multiple choice questionsA test of multiple choice questions

Pulling a block out of a bag that Pulling a block out of a bag that contains 5 red blocks and 7 blue blocks contains 5 red blocks and 7 blue blocks

Are all binomial probability experiments Are all binomial probability experiments

Binomial Distribution Problem Binomial Distribution Problem

x

x

A coin is tossed 3 times. What is the A coin is tossed 3 times. What is the probability distribution for the experiment?probability distribution for the experiment?

a)a) Define success for this experiment and list all the Define success for this experiment and list all the possible outcomes. possible outcomes.

b)b) What is the probability of success? Probability of What is the probability of success? Probability of failure?failure?

c)c) Define the random variable.Define the random variable.

d)d) Draw a tree diagram to determine the probability Draw a tree diagram to determine the probability distribution.distribution.

e)e) Complete the probability distribution for each Complete the probability distribution for each outcome.outcome.

a) Define success for this experiment and a) Define success for this experiment and

list all the possible outcomes.list all the possible outcomes.

A success for this experiment will be getting a A success for this experiment will be getting a tailtail

There are only 2 possible outcomes for this There are only 2 possible outcomes for this experimentexperiment

You get a HeadYou get a Head

oror

You get a TailYou get a Tail

b) What is the probability of success? b) What is the probability of success? Probability of failure?Probability of failure?

1( )

21

( )2

P success

P failure

d) Draw a tree diagram to determine d) Draw a tree diagram to determine the probability distributionthe probability distribution

1

2

1

8

1

4

1

21

4

1

4

1

4

1

4

1

81

8

1

81

81

8

1

8

1

8

e) Complete the probability e) Complete the probability distribution for each outcome.distribution for each outcome.

Range of Range of XX

00 11 22 33

P(x) P(x) according according

to Tree to Tree DiagramDiagram

1

83

83

8

1

8

Probability in a Binomial DistributionProbability in a Binomial Distribution

( ) x n xn xP x C p q

p is the probability of a successp is the probability of a success

q is the probability of a failureq is the probability of a failure

n is the number of trialsn is the number of trials

x is the outcomex is the outcome

Lets look back at our Lets look back at our Example….Can we find the P(x) Example….Can we find the P(x) using our Binomial Distribution using our Binomial Distribution

RanRange ge

of Xof X

00 11 22 33

P(x) P(x) Tree Tree DiagDiagramram

P(x) P(x) BinoBinomial mial DistrDistributiibutionon

1

83

8

3

81

8

0 3 0

3 0

( )

1 1( )

2 2

1( ) (1)(1)

8

1( )

8

x n xn xP x C p q

P x C

P x

P x

1 3 1

3 1

( )

1 1( )

2 2

1 1( ) (3)( )

2 8

3( )

8

x n xn xP x C p q

P x C

P x

P x

2 3 2

3 2

( )

1 1( )

2 2

1 1( ) (3)

4 2

3( )

8

x n xn xP x C p q

P x C

P x

P x

1

8

Handout Handout Question!!!!Question!!!!

Expectation for Binomial DistributionExpectation for Binomial Distribution

( )E X npSince the probability for all n Since the probability for all n trials is the same the expectation trials is the same the expectation for a success in any trial is p. for a success in any trial is p.

Homework!Homework!

Pg 385 # 1, 3 5,7,10 Pg 385 # 1, 3 5,7,10