Transcript The Binomial Distribution

Find the exact 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


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)

There must be a
fixed number of trials

The outcomes of
each trial must be
independent of
each other

The probability of a
success must remain
the same for each
trial
› 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
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


In a binomial
experiment, the
probability of
EXACTLY x successes
in n trials is:
n!
x n x
P( x) 
p q
(n  x)! x!
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.ed
u/academics/divisions/
math/lralston/Probability
%20Distributions%20-Calculator%20Instruction
s.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(

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

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?

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?
Standard Deviation
Mean
m =n * p
  n pq
Minimum

m – 2

Maximum
Mean – 2(standard deviation)
Mean + 2(standard deviation)
m + 2

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?

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?

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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