8.1 Power Point

8/7/2019 8.1 Power Point

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The BinomialThe Binomial

DistributionsDistributions


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Many random variables seen in practice amount toMany random variables seen in practice amount to

counting the number of successes incounting the number of successes in nnindependentindependent

observations, such as:observations, such as:

The number of doubles in four rolls of a pair ofThe number of doubles in four rolls of a pair of

dice.dice.

The number of patients with type A blood in aThe number of patients with type A blood in a

random sample of 10 patients.random sample of 10 patients.

The number of defective items in a sample of 20The number of defective items in a sample of 20

items.items.


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These situations are called binomial becauseThese situations are called binomial becauseeach trial has two possible outcomes:each trial has two possible outcomes:

success or failuresuccess or failure


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So Binomial SettingsSo Binomial Settings

BB: are: are bbinomialinomial meaning that they have only 2meaning that they have only 2outcomes (a success or a failure)outcomes (a success or a failure)

II: their observations are: their observations are iindependent of eachndependent of each

otherother (that is, knowing the result of one(that is, knowing the result of oneobservation tells you nothing about the otherobservation tells you nothing about the other

observations)observations)

NN: there are: there are nnobservationsobservations ((nnis a fixed number)is a fixed number)

SS: the probability of a: the probability of a ssuccess,uccess, pp, is the, is the same forame for

each observationeach observation


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Youve Got to Look Before You Leap!Youve Got to Look Before You Leap!

If you are presented with a randomIf you are presented with a random

phenomenon, it is important to be able tophenomenon, it is important to be able to

recognize it as a binomial setting or a geometricrecognize it as a binomial setting or a geometricsetting (taught in the next section.)setting (taught in the next section.)


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Binomial or Not?Binomial or Not?

The pool of potential jurors for a murder caseThe pool of potential jurors for a murder casecontains 100 people chosen at random from thecontains 100 people chosen at random from theadult residents of a large city. Each person in theadult residents of a large city. Each person in the

pool is asked whether he or she opposes the deathpool is asked whether he or she opposes the deathpenalty;penalty; XXis the number who say Yes.is the number who say Yes.

At peak periods, 15% of attempted logAt peak periods, 15% of attempted log--ins to anins to an

email service fail. Logemail service fail. Log--in attempts are independentin attempts are independentand each has the same probability of failing. Darciand each has the same probability of failing. Darcilog in repeatedly until she succeeds.log in repeatedly until she succeeds.XX is theis thenumber of lognumber of log--in attempts it took to get through.in attempts it took to get through.


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How Do We Find the Probability?How Do We Find the Probability?

Each child born to a particular set of parents hasEach child born to a particular set of parents has

probability 0.25 of having blood type O. If theseprobability 0.25 of having blood type O. If these

parents have 5 children, what is the probabilityparents have 5 children, what is the probabilitythat exactly 2 of them have type O blood?that exactly 2 of them have type O blood?


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So the Binomial Distribution isSo the Binomial Distribution is

The distribution of a random variableThe distribution of a random variable XXthat counts the number of successes is athat counts the number of successes is a

binomial distribution. Furthermore, thebinomial distribution. Furthermore, theprobability that you get exactlyprobability that you get exactlyX= kX= ksuccesses is:successes is:

and:and:

knk

ppk

n

kXP

!! )1()(!

!( - )!

! !

n k

n nC

k k n k


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Lets try:Lets try:

If 25% of all adults are college graduates, findIf 25% of all adults are college graduates, findthe probability that 3 out of an SRS of 4 adultsthe probability that 3 out of an SRS of 4 adults

are all college graduates.are all college graduates.


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With the CalculatorWith the Calculator

0:binompdf(0:binompdf(nn,,pp,, XX))

nn= number of observations= number of observations

pp = probability of the outcome= probability of the outcome

XX= random variable= random variable


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Try, Try Again!Try, Try Again!

What if we want the probability that 2 or fewerWhat if we want the probability that 2 or fewer

of our sample of 4 are college graduates?of our sample of 4 are college graduates?

Is this the same as the probability that less thanIs this the same as the probability that less than2 are college graduates?2 are college graduates?


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The Calculator AgainThe Calculator Again

A:binomcdf(A:binomcdf(nn,, pp,, XX))

nn= number of observations= number of observations

pp = probability of the outcome= probability of the outcome

XX= random variable= random variable

* This will ONLY give probabilities less than* This will ONLY give probabilities less thana specified value!!!a specified value!!!


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Mean and Standard Deviation of aMean and Standard Deviation of a

Binomial DistributionBinomial Distribution

The Mean:The Mean:

The Standard Deviation:The Standard Deviation:

Q !x np

(1 )W ! x

np p


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Median USHousehold IncomeMedian USHousehold Income

The median annual household income in the USThe median annual household income in the US

is about $39,000.is about $39,000.

Among 5 randomly selected households findAmong 5 randomly selected households find

the probability that 4 or more have incomesthe probability that 4 or more have incomes

below $39,000 per year.below $39,000 per year.


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Now consider a random sample of 16 households:Now consider a random sample of 16 households:

What is the expected number of householdsWhat is the expected number of householdswith income below $39,000?with income below $39,000?

What is the standard deviation of the number ofWhat is the standard deviation of the number of

households with incomes under $39,000?households with incomes under $39,000?

What is the probability of seeing at least 10 ofWhat is the probability of seeing at least 10 of

the 16 households with incomes under $39,000the 16 households with incomes under $39,000

annually?annually?


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Suppose in a sample of 16 US households,Suppose in a sample of 16 US households,

none had incomes below $39,000. Whatnone had incomes below $39,000. Whatmight you suspect about this sample?might you suspect about this sample?


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Binomial Distributions for various valuesBinomial Distributions for various values

ofofnnandandpp


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Is That Practical?Is That Practical?

WhenWhen nnbecomes very large, these formulas becomebecomes very large, these formulas becomecumbersome, even with a calculator.cumbersome, even with a calculator.

Lucky for us, asLucky for us, as nnincreases, the binomial distributionincreases, the binomial distributionbecomes more and more Normal.becomes more and more Normal.

So whenSo whennn

is very large, we can use the Normalis very large, we can use the Normalprobability calculation to approximate the binomial.probability calculation to approximate the binomial.We will still use the binomial formulas for mean andWe will still use the binomial formulas for mean andstandard deviation though.standard deviation though.


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Tell Me WhenTell Me When

When isWhen is nnlarge enough?large enough?

As a rule of thumb,As a rule of thumb, nnis large enough to use theis large enough to use theNormal approximation when:Normal approximation when:

npnp 10 and 10 and nn(1(1 pp) 10) 10

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