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ELEMENTARY Section 4-3 STATISTICS Binomial Probability Distributions EIGHTH Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION MARIO F. TRIOLA 1 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 2001. Addison Wesley Longman 2 Notation for Binomial Probability Distributions P(x) = probability of getting exactly x success among n trials 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 ) 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 2001. Addison Wesley Longman 3 Method 1 Binomial Probability Formula P(x) = n! • (n - x )! x! P(x) = nCx • px px • • n-x q qn-x for calculators with nCr function, r = x Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4 Method 1 – Using a formula Binomial Probability Formula P(x) = n! • (n - x )! x! Number of outcomes with exactly x successes among n trials px • n-x q Probability of x successes among n trials for any one particular order Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Method 1 – Using a formula Example: Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Method 1 – Using a formula Example: Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time. 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) = 5C3 3 2 • 0.9 • 0.1 = 0.0729 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7 For n = 15 and p = 0.10 Method 2 Table A-1 Binomial Probability Distribution n x P(x) x P(x) 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.206 0.343 0.267 0.129 0.043 0.010 0.002 0.0+ 0.0+ 0.0+ 0.0+ 0.0+ 0.0+ 0.0+ 0.0+ 0.0+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.206 0.343 0.267 0.129 0.043 0.010 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8 Method 2 – Using a table Example: Using Table A-1 for n = 15 and p = 0.10, find the following: a) The probability of exactly 3 successes b) The probability of at most 3 successes a) P(3) = 0.129 b) P(at most 3) = P(0 or 1 or 2 or 3) = P(2) or P(1) or P(2) or P(3) = 0.206 + 0.343 + 0.267 + 0.129 = 0.945 Note = This method is limited because a table may not be available for every n and/or p. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9 Method 3 – Using TI-83/4 Probabilities with “Exact” successes • Press 2nd, VARS (DISTR). • Select the option binompdf(). • Complete the entry binompdf(n, p, x) to obtain P(x). – n is the number of trials – p is the probability of success – x is the EXACT number of successes. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 Method 3 - Using TI-83/4 Probabilities with “Exact” successes • Example: What is the probability of getting exactly 2 heads when 4 tosses are made? • Solution: –P(2) = binompdf(4, 0.5, 2) –P(2) = 0.375 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11 Method 3 - Using TI-83/4 Probabilities with “At most” successes • Example: What is the probability of getting at most 2 heads when 4 tosses are made? • Express at most 2 as an inequality. – P( x ≤ 2) which means x = 0 or 1 or 2 • Solution: – P( x ≤ 2) = P(0) + P(1) + P(2) – P( x ≤ 2) = 0.0625 + 0.25 + 0.375 = 0.6875 – Where the probabilities would computed using binompdf(4,0.5, 0) then binompdf(4,0.5, 1) etc… Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12 Method 3 - Using TI-83/4 Probabilities with “At most” successes • Press 2nd, VARS, select the option binomcdf(). • Note: The “c” indicates this is a cumulative function. It adds all the probabilities from zero up to x number of successes. • Complete the entry to obtain P(At most x) = binomcdf(n, p, x), where x is the MAXIMUM number of successes. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13 Method 3 - Using TI-83/4 • Example: What is the probability of getting at most 2 heads when 4 tosses are made? • Solution: – P( x ≤ 2) = binomcdf(4, 0.5, 2) = 0.6875. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14 Method 3 - Using TI-83/4 Probabilities with “At least” successes • When doing at least problems we must use the complement rule P(A) = 1 – P(not A) • Complete the entry P(At least x) = 1 - binomcdf(n, p, x- 1), where x is the MINIMUM number of successes. Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15 Method 3 - Using TI-83/4 • Example: What is the probability of getting at least 3 heads when 4 tosses are made? • Solution: – P(x≥3) = 1 – P(x ≤ 2) – P(x≥3) = 1 - binomcdf(4, 0.5, 2) = 0.3125. Note: This is the same as • P( x ≥ 3)= P(x=3)+ P(x=4) • P( x ≥ 3)= 0.25 + 0.0625 = 0.3125 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16 Recap • P(x) Ask to find the probability of EXACT number of successes. – Formula: P(x) = nCx· px · qn-x – Calculator: P(x) = binompdf(n,p,x) • P(X x) Ask to find the probability of AT MOST a number of successes. – Calculator: P(X x ) = binomcdf(n, p, x) • P(X x) Ask to find the probability of AT LEAST a number of successes. – Calculator: P(X x ) = 1 - binomcdf(n, p, x-1) Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17