Chapter 4

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					                         Section 4.2

                       Binomial Distributions




Larson/Farber 4th ed                            26
                       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

Larson/Farber 4th ed                                         27
                       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.

Larson/Farber 4th ed                                              28
               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.



Larson/Farber 4th ed                                                  29
                  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.
   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.




Larson/Farber 4th ed                                               30
      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              31
                  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)




Larson/Farber 4th ed                                              32
                       Binomial Probability Formula

    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


Larson/Farber 4th ed                                          35
      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




Larson/Farber 4th ed                                                36
Solution: Finding Binomial Probabilities

Method 1: Draw a tree diagram and use the
Multiplication Rule




                                            37
      Solution: Finding Binomial Probabilities

    Method 2: Binomial Probability Formula




                        = 0.0486

Larson/Farber 4th ed                             38
                 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
                   x     0       1       2       3      ...
                 P(x)   0.283   0.377   0.230   0.085   ...

            Use binomial probability formula to find
             probabilities.

Larson/Farber 4th ed                                          39
              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+.

   •Construct a binomial distribution




Larson/Farber 4th ed                                           40
     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

  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



                                                             41
               Solution: Constructing a Binomial
                          Distribution

             x         P(x)
             0         0.0916   All of the probabilities are between
                                0 and 1 and the sum of the
             1         0.2808
                                probabilities is 0.9999 ≈ 1.
             2         0.3441
             3         0.2109
             4         0.0646
             5         0.0079
                       0.9999


Larson/Farber 4th ed                                                   42
      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.



 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).

Larson/Farber 4th ed                                             43
      Solution: Finding Binomial Probabilities

      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




Larson/Farber 4th ed                                            44
      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+?



 Solution:
 • Binomial with n = 138, p =
   0.38, q=0.62, x = 23
Larson/Farber 4th ed                                      45
 Example: Finding Binomial Probabilities
             Using a Table
# 26 on page 218 of the book
    x       Probability         x     Probability
        0                           0     0.237304688
        1                           1     0.395507813
        2                           2     0.263671875
        3                           3     0.087890625
        4                           4     0.014648438
        5                           5     0.000976563

Solution:
• Binomial: n = 5, p = 0.25, q = 0.75, x = 0,1,2,3,4,5
                                                         47
      Mean, Variance, and Standard Deviation

    • Mean: μ = np

    • Variance: σ2 = npq

    • Standard Deviation:




Larson/Farber 4th ed                           51
  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.


Solution: n = 12, p = 0.14, q = 0.86
 Mean: μ = np = (12)∙(0.14) = 1.68
 Variance: σ2 = npq = (12)∙(0.14)∙(0.86) ≈ 1.45
 Standard Deviation:

                                                          52
                       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

Larson/Farber 4th ed                                         54

				
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