The Binomial Distribution Section 5-4

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

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)

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, p=P(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

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

Binomial Probability Formula In a binomial experiment, the probability of EXACTLY x successes in n trials is:

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/academics/divisions/math/lralston/Probability%20Distributions%20--Calculator%20Instructions.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(

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

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?

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 doesn’t 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?

Mean & Standard Deviation Formulas m =n * p Standard Deviation

m + 2s Usual Values Mean – 2(standard deviation) m – 2s Minimum Mean – 2(standard deviation) m – 2s Maximum Mean + 2(standard deviation) m + 2s

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?

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?

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?

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