S160 #9. The Binomial Distribution, Part 1. JC Wang. February 16, 2016

Transcription

1 S160 #9 The Binomial Distribution, Part 1 JC Wang February 16, 2016

2 Outline 1 The Binomial Distribution Binomial Random Variables 2 Using Formula JC Wang (WMU) S160 #9 S160, Lecture 9 2 / 11

3 Binomial Process and Binomial Random Variable A sequence of (fixed) n observations is called a binomial process if each observation results in exactly one of two possible outcomes (conveniently called success and failure) P(success) = p and P(failure) = q = 1 p for all observations observations are independent X = total number of successes among the n observations is a binomial random variable with parameters n and p and is denoted X binomial(n, p) JC Wang (WMU) S160 #9 S160, Lecture 9 3 / 11

4 Binomial Process and Binomial Random Variable A sequence of (fixed) n observations is called a binomial process if each observation results in exactly one of two possible outcomes (conveniently called success and failure) P(success) = p and P(failure) = q = 1 p for all observations observations are independent X = total number of successes among the n observations is a binomial random variable with parameters n and p and is denoted X binomial(n, p) JC Wang (WMU) S160 #9 S160, Lecture 9 3 / 11

5 Binomial Process and Binomial Random Variable A sequence of (fixed) n observations is called a binomial process if each observation results in exactly one of two possible outcomes (conveniently called success and failure) P(success) = p and P(failure) = q = 1 p for all observations observations are independent X = total number of successes among the n observations is a binomial random variable with parameters n and p and is denoted X binomial(n, p) JC Wang (WMU) S160 #9 S160, Lecture 9 3 / 11

6 Examples of Binomial Random Variables 1 A 5-question multiple-choice quiz has 5 choices on each question. X = number of correct answers (success = correct) in the quiz by guessing all. Then X binomial(n = 5, p = 1 5 = 0.2). 2 Past experience: 40% phone respondents agree to be interviewed (success = a respondent agrees to be interviewed) for market research survey. Of 50 reached by Reliable Research, X respondents agree to be interviewed. Then X binomial(n = 50, p = 0.40). 3 Historical data shows that 20% of TV buyers at TV World purchase extended warranty (success = a buyer purchases extended warranty). X extended warranties were sold along with the 300 TV sets sold last quarter. Then X binomial(n = 300, p = 0.20). JC Wang (WMU) S160 #9 S160, Lecture 9 4 / 11

7 Examples of Binomial Random Variables 1 A 5-question multiple-choice quiz has 5 choices on each question. X = number of correct answers (success = correct) in the quiz by guessing all. Then X binomial(n = 5, p = 1 5 = 0.2). 2 Past experience: 40% phone respondents agree to be interviewed (success = a respondent agrees to be interviewed) for market research survey. Of 50 reached by Reliable Research, X respondents agree to be interviewed. Then X binomial(n = 50, p = 0.40). 3 Historical data shows that 20% of TV buyers at TV World purchase extended warranty (success = a buyer purchases extended warranty). X extended warranties were sold along with the 300 TV sets sold last quarter. Then X binomial(n = 300, p = 0.20). JC Wang (WMU) S160 #9 S160, Lecture 9 4 / 11

8 Examples of Binomial Random Variables 1 A 5-question multiple-choice quiz has 5 choices on each question. X = number of correct answers (success = correct) in the quiz by guessing all. Then X binomial(n = 5, p = 1 5 = 0.2). 2 Past experience: 40% phone respondents agree to be interviewed (success = a respondent agrees to be interviewed) for market research survey. Of 50 reached by Reliable Research, X respondents agree to be interviewed. Then X binomial(n = 50, p = 0.40). 3 Historical data shows that 20% of TV buyers at TV World purchase extended warranty (success = a buyer purchases extended warranty). X extended warranties were sold along with the 300 TV sets sold last quarter. Then X binomial(n = 300, p = 0.20). JC Wang (WMU) S160 #9 S160, Lecture 9 4 / 11

9 Examples of Binomial Random Variables 1 A 5-question multiple-choice quiz has 5 choices on each question. X = number of correct answers (success = correct) in the quiz by guessing all. Then X binomial(n = 5, p = 1 5 = 0.2). 2 Past experience: 40% phone respondents agree to be interviewed (success = a respondent agrees to be interviewed) for market research survey. Of 50 reached by Reliable Research, X respondents agree to be interviewed. Then X binomial(n = 50, p = 0.40). 3 Historical data shows that 20% of TV buyers at TV World purchase extended warranty (success = a buyer purchases extended warranty). X extended warranties were sold along with the 300 TV sets sold last quarter. Then X binomial(n = 300, p = 0.20). JC Wang (WMU) S160 #9 S160, Lecture 9 4 / 11

10 iclicker Question 9.1 The probability that a defective item is observed at a production line is A quality engineer, working at the production line, inspect an item. What is the chance that the item is found to be non-defective? A B. 1 C D E. none of the previous JC Wang (WMU) S160 #9 S160, Lecture 9 5 / 11

11 Outline 1 The Binomial Distribution Binomial Random Variables 2 Using Formula JC Wang (WMU) S160 #9 S160, Lecture 9 6 / 11

12 Using Formula to Compute Binomial Probability X binomial(n, p) P(X = j) = n! j!(n j)! pj q n j, j = 0, 1, 2,..., n. where n! = n (n 1) 2 1 and 0! = 1. Multiple-choice quiz: X binomial(5, 0.2), for example P(X = 2) = = 5! 2!(5 2)! = 5! 2! 3! (2 1) (3 2 1) = =.2048 JC Wang (WMU) S160 #9 S160, Lecture 9 7 / 11

13 TV+More Example, continued Recall that historical data shows that 20% (i.e., p = 0.2) of TV buyers at TV World purchase extended warranty. If (n =) 10 TV sets were sold in one day, what is the probability that (j =) 3 extended warranties were sold? Now, X, the number of extended waranties sold along with 10 TV sets has X binomial(10,.2) distribution and hence P(X = 3) = 10! 3!(10 3)! (0.2)3 (1 0.2) 10 3 = ! ! (0.2)3 (0.8) 7 = JC Wang (WMU) S160 #9 S160, Lecture 9 8 / 11

14 Using Formula olympics swimmer example A swimmer competes in three events in the Summer Olympics. The swimmer s winning/losing one event is independent of her result in any other event. If the probability of winning any one event is 0.45, what is the chance that she wins two or three events? X binomial(3, 0.45) 3! 2!1! ! }{{} 3!0! = = }{{} P(X = 2) P(X = 3) JC Wang (WMU) S160 #9 S160, Lecture 9 9 / 11

15 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

16 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

17 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

18 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

19 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

20 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

21 The Language of Probability Note first that X, the number of successes, can only assume values 0, 1,..., n. only 2 or exactly 2 : P(X = 2) at most 3 or no more than 3 or 3 or less : P(X 3) = P(X = 0, 1, 2, or 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). at least 5 or no less than 5 or 5 or more if n = 10: P(X 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) at least 8 or no less than 8 or 8 or more if n = 10: P(X 8) = P(X = 8) + P(X = 9) + P(X = 10) etc. JC Wang (WMU) S160 #9 S160, Lecture 9 10 / 11

22 iclicker Question 9.2 The probability that a defective item is observed at a production line is A quality engineer, working at the production line, inspect the next 4 items. What is the set of possible number of defectives? A. {1,2,3,4} B. {0,1,2,3,4} C. {1,2} D. {3,4} E. none of the previous JC Wang (WMU) S160 #9 S160, Lecture 9 11 / 11