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# ch17 - STA 2023 Holbrook Probability Models 1 Discrete...

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Unformatted text preview: STA 2023 - Holbrook Probability Models Chapter 17 17 - 1 Discrete Probability Distribution Function STA 2023 - Holbrook 1. Type of Model Representation of Some Representation Underlying phenomenon Underlying 2. Mathematical Formula Mathematical 3. Represents Discrete 3. Represents Random Variable Random 4. Used to Get Probabilities Used for each X value for 17 - 2 P (X = x ) x λe x! -λ = STA 2023 - Holbrook Discrete Probability Distribution Models Discrete Probability Distribution Binomial Poisson Geometric 17 - 3 STA 2023 - Holbrook Skip: Page 433­434, The Geometric Model (Pages 317­318 First Edition) Skip: Page 441­443 The Poisson Model 17 - 4 STA 2023 - Holbrook Binomial Distribution 17 - 5 Binomial Distribution Properties STA 2023 - Holbrook 1. Sequence of n Identical Trials Sequence 2. Each Trial Has 2 Outcomes ‘Success’ (Desired Outcome) or ‘Failure’ 3. Probability of Success Remains Probability Constant from Trial to Trial Constant 4. Trials Are Independent Trials 17 - 6 Binomial Distribution STA 2023 - Holbrook 1. Number of ‘Successes’ in a Sample of Number Sample n Observations (Trials) Observations # Reds in 15 Spins of Roulette Wheel # Defective Items in a Batch of 5 Items # Correct on a 33 Question Exam # Customers Who Purchase Out of 100 Customers Customers Who Enter Store Customers 17 - 7 STA 2023 - Holbrook Binomial Probability Distribution Function P ( X = x) =n C x p q x n −x n! = p x (1 − p ) n −x x!( n − x )! P(X = x) = Probability of x ‘Successes’ (X Probability ‘Successes’ n = Sample Size (or Number of Trials) Number p = Probability of ‘Success’ x = Number of ‘Successes’ in Sample Number Sample (x = 0, 1, 2, ..., n) 17 - 8 STA 2023 - Holbrook Binomial Probability Distribution Example Experiment: Toss 1 Coin 5 Times in a Row. Experiment: Note # Tails. What’s the Probability of 3 Tails? Note n! x n −x p( X = x) = p (1 − p ) x!( n − x )! 5! p ( X = 3) = .53 (1 −.5)5−3 3!(5 − 3)! 17 - 9 = 0.3125 STA 2023 - Holbrook Same Problem with TI­83 17 - 10 10 STA 2023 - Holbrook Binomial Probability Distribution Example Experiment: Toss 1 Coin 5 Times in a Row. Experiment: Note # Tails. What’s the Probability of 3 Tails? Note • 2nd “VARS” (pulls up the “DISTR” menu) • Scroll down to “0” ( “binompdf(“ ) Scroll • Hit Enter Hit • binompdf(number of trials, p, x) binompdf(number • binompdf(5, .5, 3) binompdf(5, • = 0.3125 0.3125 17 - 11 11 STA 2023 - Holbrook Binomial Probability Distribution Example Experiment: Toss 1 Coin 5 Times in a Row. Experiment: Note # Tails. What’s the Probability of 3 Tails? Note • Create a table of all possible x, p(x) pairs. Create 17 - 12 12 STA 2023 - Holbrook Binomial Distribution Characteristics Mean µ = E ( x ) = np .6 .4 .2 .0 .6 .4 .2 .0 P(X) 1 2 3 4 5 n = 5 p = 0.5 X 0 17 - 13 13 n = 5 p = 0.1 X 0 Standard Deviation σ = np (1 − p ) P(X) 1 2 3 4 5 STA 2023 - Holbrook Binomial Distribution Thinking Challenge You’re a telemarketer selling You’re service contracts for Macy’s. You’ve sold 20 in your last 100 calls (p = .20). If you 100 ). call 12 people tonight, 12 what’s the probability of what’s A. B. C. C. D. D. No sales? Exactly 2 sales? At most 2 sales? At least 2 sales? 17 - 14 14 STA 2023 - Holbrook Binomial Distribution Solution* Using the binompdf function on our TI-83: A. p(X=0) = .069 [ binompdf(12,.2,0) ] (X=0) .069 B. p(X=2) = .283 [ binompdf(12,.2,2) ] (X=2) binompdf(12,.2,2) C. p(at most 2) = p(X=0) + p(X=1) + p(X=2) (X=0) (X=1) = .069 + .206 + .283 = .558 .558 D. p(at least 2) = p(X=2) + p(X=3)...+ p(X=12) (X=2) (X=3)...+ = 1 - [p(X=0) + p(X=1)] (X=0) (X=1)] = 1 - .069 - .206 .069 = .725 .725 17 - 15 15 STA 2023 - Holbrook Binomial Distribution Thinking Challenge You’re a telemarketer selling You’re service contracts for Macy’s. You’ve sold 20 in your last 100 calls (p = .20). Find the 100 ). mean and standard deviation. mean 17 - 16 16 STA 2023 - Holbrook Binomial Distribution Solution* µ = E ( x) = np = (12)(0.2) = 2.4 σ = np(1 − p ) = (12)(0.2)(1 − 0.2) = 1.386 17 - 17 17 End of Chapter Any blank slides that follow are blank intentionally. ...
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