Chapter 6: Binomial Probability Distributions

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Jun 10, 2022 Chapter 6: Chapter 6: Binomial Probability Binomial Probability Distributions Distributions

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Chapter 6: Binomial Probability Distributions. In Chapter 6:. 6.1 Binomial Random Variables 6.2 Calculating Binomial Probabilities 6.3 Cumulative Probabilities 6.4 Probability Calculators 6.5 Expected Value and Variance of Binomial Random Variables - PowerPoint PPT Presentation

Transcript of Chapter 6: Binomial Probability Distributions

Page 1: Chapter 6:  Binomial Probability Distributions

Apr 20, 2023

Chapter 6: Chapter 6: Binomial Probability Binomial Probability

DistributionsDistributions

Page 2: Chapter 6:  Binomial Probability Distributions

In Chapter 6:

6.1 Binomial Random Variables6.2 Calculating Binomial Probabilities6.3 Cumulative Probabilities6.4 Probability Calculators6.5 Expected Value and Variance of Binomial Random Variables6.6 Using the Binomial Distribution to Help Make Judgments

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§6.1 Binomial Random Variables• Binomial = a family of discrete random

variables

• Binomial random variable ≡ the random number of successes in n independent Bernoulli trials (a Bernoulli trials has two possible outcomes: “success” or “failure”

• Binomials random variables have two parametersn number of trialsp probability of success of each trial

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Binomial Example• Consider the random number of successful

treatments when treating four patients

• Suppose the probability of success in each instance is 75%

• The random number of successes can vary from 0 to 4

• The random number of successes is a binomial with parameters n = 4 and p = 0.75

• Notation: Let X ~b(n,p) represent a binomial random variable with parameters n and p. The illustration variable is X ~ b(4, .75)

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§6.2 Calculating Binomial Probabilities

xnxxn qpCxX )Pr(

where

nCx ≡ the binomial coefficient (next slide)

p ≡ probability of success for each trial

q ≡ probability of failure = 1 – p

Formula for binomial probabilities:

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Binomial Coefficient

)!(!

!

xnx

nCxn

where ! represents the factorial function, calculated:x! = x (x – 1) (x – 2) … 1For example, 4! = 4 3 2 1 = 24By definition 1! = 1 and 0! = 1

6)12)(12(

1234

)!2)(!2(

!4

)!24)(!2(

!424

C

For example:

Formula for the binomial coefficient:

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Binomial Coefficient

)!(!

!

xnx

nCxn

The binomial coefficient is called the “choose function” because it tells you the number of ways you could choose x items out of n

nCx the number of ways to choose x items out of n

For example, 4C2 = 6 means there are six ways to choose two items out of four

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Binomial Calculation – Example Recall the “Four patients example”. Four patients; probability of success of each treatment = .75. The number of success is the binomial random variable X ~ b(4,.75). Note q = 1 −.75 = .25. What is the probability of observing 0 successes under these circumstances?

0039.

0039.11

250750!4!0

!4

250750

)0(Pr

40

04004

..

..C

qpCX xnxxn

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X~b(4,0.75), continued

Pr(X = 1) = 4C1 · 0.751 · 0.254–1

= 4 · 0.75 · 0.0156

= 0.0469

Pr(X = 2) = 4C2 · 0.752 · 0.254–2

= 6 · 0.5625 · 0.0625

= 0.2106

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X~b(4, 0.75) continued

Pr(X = 3) = 4C3 · 0.753 · 0.254–3

= 4 · 0.4219 · 0.25

= 0.4219

Pr(X = 4) = 4C4 · 0.754 · 0.254–4

= 1 · 0.3164 · 1

= 0.3164

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pmf for X~b(4, 0.75)Tabular and graphical forms

x Pr(X = x)

0 0.0039

1 0.0469

2 0.2109

3 0.4210

4 0.3164

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Area Under The Curve

Pr(

X =

2)

=.2

109 ×

1.

0

Recall the area under the curve (AUC) concept.

AUC = probability!

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§6.3: Cumulative Probability

• Recall the cumulative probability concept

• Cumulative probability ≡ the probability of that value or less

• Pr(X x) • Corresponds to left

tail of pmf

Pr(X 2) on X ~b(4,.75)

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Cumulative Probability Function

• Cumulative probability function lists cumulative probabilities for all possible outcome

• Example: The cumulative probability function for X~b(4, 0.75)Pr(X 0) = 0.0039

Pr(X 1) = 0.0508

Pr(X 2) = 0.2617

Pr(X 3) = 0.6836

Pr(X 4) = 1.0000

Pr(X 1) = Pr(X = 0) + Pr(X = 1) = .0039 + .0469 = 0.0508

Pr(X 2) = Pr(X = 0) + Pr(X = 1) + Pr(X = 2)

Pr(X 4) = Pr(X = 0) + Pr(X = 1) + Pr(X = 2) + Pr(X = 3)

Pr(X 4) = Pr(X = 0) + Pr(X = 1) + … + Pr(X = 4)

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§6.5: Expected Value and Variance for Binomials

• The expected value (mean) μ of a binomial pmf is its “balancing point”

• The variance σ2 is its spread

• Shortcut formulas:

np npq2

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Expected Value and Variance, Binomials, Illustration

For the “Four patients” pmf of X~b(4,.75)

μ = n∙p = (4)(.75) = 3

σ2 = n∙p∙q = (4)(.75)(.25) = 0.75

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§6.6 Using the Binomial• Suppose we observe 2

successes in the “Four patients” example

• Note μ = 3, suggesting we should see 3 success on average

• Does the observation of 2 successes cast doubt on p = 0.75?

• No, because Pr(X 2) = 0.2617 is not too unusual

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StaTable Probability Calculator• Calculates

probabilities for many types of random variables

• This figure shows probabilities for X~b(4,0.75)

• Available in Java, Windows, and Palm versions (download from website)

Pr(X = 2) = .2109

Pr(X ≤ 2) = .2617

x = 2

p = .75

n = 4