Post on 11-Jan-2016
Section 4.2
Binomial Distributions
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Section 4.2 Objectives
• Determine if a probability experiment is a binomial experiment
• Find binomial probabilities using the binomial probability formula
• Find binomial probabilities using technology and a binomial table
• Graph a binomial distribution• Find the mean, variance, and standard deviation of a
binomial probability distribution
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Binomial Experiments
1. The experiment is repeated for a fixed number of trials, where each trial is independent of other trials.
2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F).
3. The probability of a success P(S) is the same for each trial.
4. The random variable x counts the number of successful trials.
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Notation for Binomial Experiments
Symbol Description
n The number of times a trial is repeated
p = P(S) The probability of success in a single trial
q = P(F) The probability of failure in a single trial (q = 1 – p)
x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, … , n.
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Example: Binomial Experiments
Decide whether the experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x.
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1. Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 12 adults and ask each to name his or her favorite cookie.
Solution: Binomial Experiments
Binomial Experiment
1. Each question represents a trial. There are 12 adults questioned, and each one is independent of the others.
2. There are only two possible outcomes of interest for the question: Oatmeal Raisin (S) or not Oatmeal Raisin (F).
3. The probability of a success, P(S), is 0.10, for oatmeal raisin.
4. The random variable x counts the number of successes - favorite cookie is Oatmeal raisin.
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Solution: Binomial Experiments
Binomial Experiment• n = 12 (number of trials)• p = 0.10 (probability of success)• q = 1 – p = 1 – 0.10 = 0.90 (probability of failure)• x = 0, 1, 2, 3, 4, 5, 6, 7, 8 (number of people that like
oatmeal raisin cookies)
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Binomial Probability Formula
35Larson/Farber 4th ed
Binomial Probability Formula• The probability of exactly x successes in n trials is
• n = number of trials• p = probability of success• q = 1 – p probability of failure• x = number of successes in n trials
Example: Finding Binomial Probabilities
Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 4 adults and ask each to name his or her favorite cookie.
Find the probability that the number who say oatmeal raisin is their favorite cookie is (a) exactly 2, (b) at least 1 and (c) less than four
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Solution: Finding Binomial Probabilities
Method 1: Draw a tree diagram and use the Multiplication Rule
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Solution: Finding Binomial Probabilities
Method 2: Binomial Probability Formula
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= 0.0486
Binomial Probability Distribution
Binomial Probability Distribution• List the possible values of x with the corresponding
probability of each.• Example: Binomial probability distribution for
Oatmeal Cookies: n = 12 , p = 0.10
Use binomial probability formula to find probabilities.
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x 0 1 2 3 ...
P(x) 0.283 0.377 0.230 0.085 ...
Example: Constructing a Binomial Distribution
Thirty eight percent of people in the United States have type O+ blood. You randomly select five Americans and ask them if their blood type is O+.
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•Construct a binomial distribution
Solution: Constructing a Binomial Distribution
• 38% of Americans have blood type O+.• n = 5, p = 0.38, q = 0.62, x = 0, 1, 2, 3, 4, 5
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P(x = 0) = 5C0(0.38)0(0.62)5 = 1(0.38)0(0.62)5 ≈ 0.0916
P(x = 1) = 5C1(0.38)1(0.62)4 = 5(0.38)1(0.62)4 ≈ 0.2807
P(x = 2) = 5C2(0.38)2(0.62)3 = 10(0.38)2(0.62)3 ≈ 0.3441
P(x = 3) = 5C3(0.38)3(0.62)2 = 10(0.38)3(0.62)2 ≈ 0.2109
P(x = 4) = 5C4(0.38)4(0.62)1 = 5(0.38)4(0.62)1 ≈ 0.0646
P(x = 5) = 5C5(0.38)5(0.62)0 = 1(0.38)5(0.62)0 ≈ 0.0079
Solution: Constructing a Binomial Distribution
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x P(x)
0 0.0916
1 0.2808
2 0.3441
3 0.2109
4 0.0646
5 0.0079
0.9999
All of the probabilities are between 0 and 1 and the sum of the probabilities is 0.9999 ≈ 1.
Example: Finding Binomial Probabilities
Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 4 adults and ask each if their favorite cookie is oatmeal raisin.
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Solution: • n = 4, p = 0.10, q = 0.90• At least two means 2 or more.• Find the sum of P(2), P(3) and P(4).
Solution: Finding Binomial Probabilities
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P(x = 2) = 4C2(0.10)2(0.90)2 = 6(0.10)2(0.90)2 ≈ 0.0486
P(x = 3) = 4C3(0.10)3(0.90)1 = 4(0.10)3(0.90)1 ≈ 0.0036
P(x = 4) = 4C4(0.10)4(0.90)0 = 1(0.10)4(0.90)0 ≈ 0.0001
P(x ≥ 2) = P(2) + P(3) + P(4) ≈ 0.0486 + 0.0036 + 0.0001 ≈ 0.0523
Example: Finding Binomial Probabilities Using Technology
Thirty eight percent of people in the United States have type O+ blood. You randomly select 138 Americans and ask them if their blood type is O+. What is the probability that exactly 23 have blood type O+?
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Solution:• Binomial with n = 138, p =
0.38, q=0.62, x = 23
Example: Finding Binomial Probabilities Using a Table
# 26 on page 218 of the book
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Solution:• Binomial: n = 5, p = 0.25, q = 0.75, x = 0,1,2,3,4,5
x Probability
0 0.237304688
1 0.395507813
2 0.263671875
3 0.087890625
4 0.014648438
5 0.000976563
x Probability
0
1
2
3
4
5
Mean, Variance, and Standard Deviation
• Mean: μ = np
• Variance: σ2 = npq
• Standard Deviation:
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Example: Finding the Mean, Variance, and Standard Deviation
Fourteen percent of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each if cashews are their favorite nut. Find the mean, variance and standard deviation.
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Solution: n = 12, p = 0.14, q = 0.86
Mean: μ = np = (12)∙(0.14) = 1.68Variance: σ2 = npq = (12)∙(0.14)∙(0.86) ≈ 1.45Standard Deviation:
Section 4.2 Summary
• Determined if a probability experiment is a binomial experiment
• Found binomial probabilities using the binomial probability formula
• Found binomial probabilities using technology and a binomial table
• Graphed a binomial distribution• Found the mean, variance, and standard deviation of
a binomial probability distribution
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