Definitions Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1 Monday, March 18, 2013
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1 Definitions Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1
2 Definitions Binomial Probability Distribution Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1
3 Definitions Binomial Probability Distribution 1. The experiment must have a fixed number of trials. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1
4 Definitions Binomial Probability Distribution 1. The experiment must have a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn t affect the probabilities in the other trials.) Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1
5 Definitions Binomial Probability Distribution 1. The experiment must have a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1
6 Definitions Binomial Probability Distribution 1. The experiment must have a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories. 4. The probabilities must remain constant for each trial. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 1
7 Notation for Binomial Probability Distributions Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
8 Notation for Binomial Probability Distributions n = fixed number of trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
9 Notation for Binomial Probability Distributions n = fixed number of trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
10 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
11 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
12 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
13 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
14 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p ) Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
15 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p ) Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
16 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p ) P(x) = probability of getting exactly x success among n trials Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
17 Notation for Binomial Probability Distributions n = fixed number of trials x = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials (q = 1 - p ) P(x) = probability of getting exactly x success among n trials Be sure that x and p both refer to the same category being called a success. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 2
18 Binomial Probability Formula P(x) = n! (n - x )! x! p x q n-x Number of outcomes with exactly x successes among n trials Probability of x successes among n trials for any one particular order Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 3
19 Binomial Probability Formula P(x) = n! (n - x )! x! p x q n-x Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 4
20 Binomial Probability Formula P(x) = n! (n - x )! x! p x q n-x P(x) = n C x p x q n-x Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 4
21 Binomial Probability Formula P(x) = n! (n - x )! x! p x q n-x P(x) = n C x p x q n-x for calculators with n C r key, where r = x Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 4
22 Example 1 Given 5 different requests from AT&T directory assistance and assuming in general, AT&T is correct 90% of the time, find: a) the probability that exactly 3 are correct This is a binomial experiment where: n = 5 x = 3 p = 0.90 q = 0.10 Using the binomial probability formula to solve: P(3) = 5 C = Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 5
23 Example 1 Given 5 different requests from AT&T directory assistance and assuming in general, AT&T is correct 90% of the time, find: b) the probability that exactly 4 are correct P(4) = 5 C = c) the probability that exactly 5 are correct P(5) = 5 C = d) the probability that at least 3 are correct P(3) + P(4) + P(5) = = Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 6
24 Example 2 It is known that 80% of the seeds of particular flowers will germinate in the right conditions. If a packet of 10 seeds is purchased, find the probability that: a) at most two will fail to germinate. b) exactly 8 will germinate. c) Between 3 and 6 seeds inclusive will germinate
25 Example 2 1. a) Number failing to germinate ~ B(10,0.2), where success is the failure to germinate P(0) + P(1) + P(2) = 10 C 0 x x C 1 x x C 2 x x = = % b) P(8) = 10 C = % c) Between 3 and 6 inclusive will germinate P(3) + P(4) + P(5) + P(6) = 10 C 3 x x C 4 x x C 5 x x C 6 x x = = %
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