**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 .. 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

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

**Pr(X = 2)** =.2109 × 1.0 AUC = probability!

**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 =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 Function Cumulative 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?