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### 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 ..

6.6 Using the Binomial Distribution to Help Make Judgments

Binomial Random Variables

- Bernoulli trial≡ a random event with two possible outcomes (“success” or “failure”)
- Binomial random variable ≡ the random number of successes in n independent Bernoulli trials, each trial with the same probability of success
- Binomials have two parameters:n number of trialsp probability of success of each trial

Binomials (cont.)

- Only two outcomes are possible (success and failure)
- The outcome of each trial does not depend on the previous trial (independence)
- The probability for success p is the same for each trial
- Trials are repeated a specified number of times n

Calculating Binomial Probabilities by hand

Formula:

where

nCx≡ the binomial coefficient (next slide)

p≡ probability of success for a single trial

q≡ probability of failure for single trial = 1 – p

Binomial Coefficient

Formula for the binomial coefficient:

where ! represents the factorial function:

x!= x (x – 1) (x – 2) … 1

For example, 4! = 4 3 2 1 = 24

By definition 1! = 1 and 0! = 1

For example:

Binomial Coefficient

The binomial coefficient tells you the number of ways you could choose x items out of n

nCx the number of ways to x items out of n

For example, 4C2 = 6 Therefore, there are 6 ways to choose 2 items out of 4.

Binomial Calculation – Example

“Four patients example”: X ~ b(4,.75).

Note q = 1 −.75 = .25.

What is the probability of 0 successes?

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

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

Cumulative Probability = Pr(X x) = Left “Tail”

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

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

Pr(X =0) + Pr(X = 1) + … + Pr(X = 3)

Pr(X =0) + Pr(X = 1) + … + Pr(X = 4)

Cumulative Probability FunctionCumulative probability function (cdf) = cumulative probabilities for all outcome

Example: cdf 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

Calculating Binomial Probabilities with the StaTable Utility

StaTable is a free computer program that calculates probabilitiesfor many types of random variables, including binomials

StaTable Binomial Calculator

Number of successes x

Binomial parameter p

Binomial parameter n

Calculates Pr(X = x)

Calculates Pr(X≤ x)

StaTable Probability Calculator

StaTable

Exact and cumulative probability of “2” for X~b(n = 4, p = .75)

x = 2

p = .75

n = 4

Pr(X = 2) = .2109

Pr(X≤ 2) = .2617

§6.5: Expected Value and Variance for Binomials

- Expected value μ
- Variance σ2
- Shortcut formulas:

Expected Value and Variance, Binomials, Illustration

For X~b(4,.75)

μ = n∙p = (4)(.75) = 3

σ2 = n∙p∙q = (4)(.75)(.25) = 0.75

§6.6 Using the Binomial

- Suppose we observe 2 successes in a “Four patients” experiment?
- Assume X~b(4, .75)
- 3 success are expected
- Does the observation of 2 successes cast doubt on p = 0.75?

Pr(X 2) = 0.2617. What does this infer?

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