1 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 4-3 Binomial Probability Distributions

2 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

3 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

4 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  P(x) = p x q n-x ( n - x ) ! x ! Binomial Probability Formula n !n ! Method 1  P(x) = n C x p x q n-x for calculators with n C r function, r = x

5 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P(x) = p x q n-x n !n ! ( n - x ) ! x ! Number of outcomes with exactly x successes among n trials Probability of x successes among n trials for any one particular order Binomial Probability Formula Method 1 – Using a formula

6 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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. Method 1 – Using a formula

7 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 = 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. 3 2 Method 1 – Using a formula

8 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P( x ) n x P( x ) x Table A-1 Binomial Probability Distribution For n = 15 and p = 0.10 Method 2

9 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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) = b) P(at most 3) = P(0 or 1 or 2 or 3) = P(2) or P(1) or P(2) or P(3) = = Note = This method is limited because a table may not be available for every n and/or p. Method 2 – Using a table

10 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probabilities with “Exact” successes Press 2 nd, 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. Method 3 – Using TI-83/4

11 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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) = Method 3 - Using TI-83/4 Probabilities with “Exact” successes

12 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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) = = –Where the probabilities would computed using binompdf(4,0.5, 0) then binompdf(4,0.5, 1) etc… Method 3 - Using TI-83/4

13 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probabilities with “At most” successes Press 2 nd, 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. Method 3 - Using TI-83/4

14 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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) = Method 3 - Using TI-83/4

15 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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. Method 3 - Using TI-83/4

16 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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) = Note: This is the same as P( x ≥ 3)= P(x=3)+ P(x=4) P( x ≥ 3)= = Method 3 - Using TI-83/4

17 Chapter 4. Section 4-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Recap P(x) Ask to find the probability of EXACT number of successes. –Formula: P(x) = n C x · p x · q n-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)