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Chapter 6: Binomial Probability Distributions

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.

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Chapter 6: Binomial Probability Distributions

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  1. Chapter 6: Binomial Probability Distributions

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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:

  7. 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.

  8. Binomial Calculation – Example “Four patients example”: X ~ b(4,.75). Note q = 1 −.75 = .25. What is the probability of 0 successes?

  9. 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

  10. 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

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

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

  13. Cumulative Probability = Pr(X  x) = Left “Tail” This figure illustrates Pr(X  2) on X ~b(4,.75)

  14. 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

  15. Calculating Binomial Probabilities with the StaTable Utility StaTable is a free computer program that calculates probabilitiesfor many types of random variables, including binomials

  16. StaTable Binomial Calculator Number of successes x Binomial parameter p Binomial parameter n Calculates Pr(X = x) Calculates Pr(X≤ x)

  17. 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

  18. §6.5: Expected Value and Variance for Binomials • Expected value μ • Variance σ2 • Shortcut formulas:

  19. 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

  20. §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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