The Binomial Distribution

1
The Binomial Distribution

• Section 5-4

2
Objectives

• Find the probability for X successes in n trials
  of a binomial experiment
• Find the mean, variance, and standard deviation
  for the variable of a binomial distribution

3
Binomial Experiments

• Situations that have only two outcomes or can be
  reduced to two outcomes
• A coin is tossed (Heads or Tails)
• A child is born (Boy or Girl)
• Outcome of a basketball game (Win or Lose)
• Answer a true/false question (True or False)
• Medical Treatment (Effective or Ineffective)
• Answer a multiple choice question (Correct or
  Incorrect)

4
Binomial Experiments must satisfy the following
requirements

• There must be a fixed number of trials
• Represented by n
• Each trial can have only two outcomes or outcomes
  that can be reduced to two outcomes
• Success, pP(S)
• Failure, q P(F) 1-p

• The outcomes of each trial must be independent of
  each other
• The probability of a success must remain the same
  for each trial

5
Binomial or Not?

• Randomly selecting 12 jurors and recording their
  nationalities
• Recording the genders of 250 newborn babies
• Determining whether each of 500 defibrillators is
  acceptable or defective
• Treating 50 smokers with Nicorette and asking
  them how their mouth and throat feel

6
Binomial Probability Formula

• In a binomial experiment, the probability of
  EXACTLY x successes in n trials is

7
Binomial Distribution

• Since computing probabilities using the formula
  can be quite tedious, we will use the TI-83/84
  calculator to help us find and interpret the
  probabilities
• Link to instructions http//www.highlands.edu/aca
  demics/divisions/math/lralston/Probability20Distr
  ibutions20--Calculator20Instructions.htm

• KEY WORDS/PHRASES
• will help to determine calculator commands
• Exactly x successes Use command binompdf(
• At most x successes
• Use command binomcdf(
• At least x successes Use command 1
  binomcdf(

8
Examples

• The CBS television show, 60 Minutes, has been
  successful for many years. That show recently
  had a share of 20, meaning that among the TV sets
  in use, 20 were tuned to 60 Minutes. Assume that
  an advertiser wants to verify that 20 share
  value by conducting its own survey. A pilot
  survey begins with 10 households having TV sets
  in use at the time of a 60 Minutes broadcast
• Find the probability that none of the households
  are tuned to 60 Minutes
• Find the probability that at least one household
  is tuned to 60 Minutes
• Find the probability that at most one household
  is tuned to 60 Minutes

9
Examples

• Air America has a policy of booking as many as 20
  persons on an airplane that can seat only 14.
  (Past studies have revealed that only 85 of the
  booked passengers actually arrive for the flight.
  Find the probability that if Air America books
  20 persons, not enough seats will be available.
  That is, find P(at least 15 persons arrive for
  flight) Is this probability low enough so that
  overbooking is not a real concern for passengers?
  

10
Example

• The Medassist Pharmaceutical Company receives
  large shipments of aspirin tablets and uses this
  acceptance sampling plan Randomly select and
  test 24 tablets, then accept the whole batch if
  there is only one or none that doesnt meet the
  required specifications. If a particular
  shipment of thousands of aspirin tablets has a 4
  rate of defects, what is the probability that
  this shipment will be accepted?

11
Mean Standard Deviation Formulas

• Mean

• m n p

• Standard Deviation

12
Usual Values

• Minimum

• Mean 2(standard deviation)
• m 2s

• Maximum

• Mean 2(standard deviation)
• m 2s

13
Examples

• Air America has a policy of booking as many as 15
  persons on an airplane that can seat only 14.
  (Past studies have revealed that only 85 of the
  booked passengers actually arrive for the flight.
  
• What is the average number of passengers on Air
  America if 15 reservations are accepted?
• What is the standard deviation?
• What is the usual minimum number of passengers
  on Air America?
• What is the usual maximum number of passengers
  on Air America?

14
Examples

• Several Psychology students are unprepared for a
  surprise true/false test with 16 questions and
  all of their answers are guesses.
• Find the mean and standard deviation for the
  number of correct answers for such students
• Would it be unusual for a student to pass by
  guessing and getting at least 10 correct answers?
  Why or why not?

15
Examples

• Several Economics students are unprepared for a
  multiple-choice quiz with 25 questions, and all
  of their answers are guesses. Each question has
  five possible answers and only one of them is
  correct.
• Find the mean and standard deviation for the
  number of correct answers for such students
• Would it be unusual for a student to pass by
  guessing and getting at least 15 correct answers?
  Why or why not?

16
Assignment

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