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CHAPTER 6
BINOMIAL DISTRIBUTION
Outline
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The context
The properties
Notation
Formula
Use of table
Use of Excel
Mean and variance
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BINOMIAL DISTRIBUTION
THE CONTEXT
• An important property of the binomial distribution:
– An outcome of an experiment is classified into one of
two mutually exclusive categories - success or failure.
• Example: Suppose that a production lot contains 100
items. The producer and a buyer agree that if at most
2 out of a sample of 10 items are defective, then all
the remaining 90 items in the production lot will be
purchased without further testing. Note that each item
can be defective or non defective which are two
mutually exclusive outcomes of testing. Given the
probability that an item is defective, what is the
probability that the 90 items will be purchased without
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further testing?
BINOMIAL DISTRIBUTION
THE CONTEXT
Trial
Flip a coin
Apply for a job
Answer a Multiple
choice question
Two Mut. Excl. and exhaustive outcomes
Head / Tail
Get the job / not get the job
Correct / Incorrect
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BINOMIAL DISTRIBUTION
THE PROPERTIES
• The binomial distribution has the following properties:
1. The experiment consists of a finite number of trials. The
number of trials is denoted by n.
2. An outcome of an experiment is classified into one of
two mutually exclusive categories - success or failure.
3. The probability of success stays the same for each trial.
The probability of success is denoted by p.
4. The trials are independent.
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BINOMIAL DISTRIBUTION
THE NOTATION
• Notation
– n : the number of trials
– x : the number of observed successes
– p : the probability of success on each trial
– q : the probability of failure on each trial
• Note:
– Since success and failure are two mutually exclusive
and exhaustive events
p+ q=1
p=1- q
q=1- p
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BINOMIAL DISTRIBUTION
THE PROBABILITY DISTRIBUTION
• The binomial probability distribution gives the probability of
getting exactly x successes out of a total of n trials.
• The probability of getting exactly x successes out of a total
of n trials is as follows:
P X  x   p x 
 C xn p x q n  x

n!  x n  x
 p q
 
 x!n  x ! 
n
• Note: In the above C x gives the number of different ways
of choosing x objects out of a total of n objects
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BINOMIAL DISTRIBUTION
THE PROBABILITY DISTRIBUTION
Example 1: If you toss a fair coin twice, what is the
probability of getting one head and one tail? From our
previous discussion, we know that the answer is 0.50.
Verify if the binomial probability distribution formula gives
the same answer.
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BINOMIAL DISTRIBUTION
THE PROBABILITY DISTRIBUTION
Example 2: If you toss a fair coin three times, what is the
probability of getting at most one head (at least two tails)?
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BINOMIAL DISTRIBUTION
THE PROBABILITY DISTRIBUTION
Example 3 (self study): Redo Example 2 with a probability
tree and verify if the probability tree gives the same
answer.
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BINOMIAL DISTRIBUTION
NECESSITY OF A TABLE OR SOFTWARE
Example 4 (do not solve): If you toss a fair coin 25 times,
what is the probability of getting at most 10 heads (at least
15 tails)? Do not solve this problem, but discuss the
computation required by the binomial probability
distribution formula.
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BINOMIAL DISTRIBUTION
USE OF TABLE
• Table 1, Appendix B, pp. 830-834 gives the probability of
getting at most k successes P X  k  out of a total of n
trials, for probability of success in each trial p.
• The table does not include the cases where k=n. The
probability of getting at most n successes out of a total of
n trials is 1.00 - so, if k=n, the required probability is 1.00
• The table can be used to find the probability of
– exactly k successes: P X  k   P X  k   P X  k 1
– at least k successes: P X  k   1  P X  k 1
– successes between a and b:
Pa  X  b  P X  b  P X  a 1
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BINOMIAL DISTRIBUTION
USE OF TABLE
Example 5: Find the following using Table 1:
P X  2 | n  5, p  0.30
P X  4 | n  9, p  0.60
Example 6: Find the following: P X  5 | n  5, p  0.30
Example 7: Find the following using Table 1:
P X  2 | n  5, p  0.30
P X  3 | n  5, p  0.30
P2  X  4 | n  5, p  0.30
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BINOMIAL DISTRIBUTION
USE OF TABLE
Example 8 (self study): If you toss a fair coin 25 times, what
is the probability of getting at most 10 heads (at least 15
tails)? Solve this problem using the Table 1.
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BINOMIAL DISTRIBUTION
USE OF EXCEL
• The Excel function BINOMDIST gives P X  k  and P X  k 
• It takes four arguments. The first 3 arguments are k,n,p
• The last one is TRUE for P X  k  and FALSE for P X  k 
Example 9 (self study): If you toss a fair coin 25 times, what
is the probability of getting at most 10 heads (at least 15
tails)? Solve this problem using Excel. Verify if Excel gives
the same answer as it is given by Table 1 in Example 8.
Answer: P X  10 | n  25, p  0.50
=BINOMDIST(10,25,0.5,TRUE)
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BINOMIAL DISTRIBUTION
MEAN AND VARIANCE
• If X is a binomial random variable, the mean and the
variance of X are:
E  X     np
V  X    2  npq
• E(X) is the mean or expected value of X
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V(X) is the variance of X
n is the number of trials
p is the probability of success on each trial
q is the probability of failure on each trial = 1-p
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BINOMIAL DISTRIBUTION
MEAN AND VARIANCE
Example 10: Let X be a random variable that gives number
of heads when a fair coin is tossed 4 times. Compute E(X)
and V(X). The interpretation of E(X) and V(X) is done in
Example 11.
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BINOMIAL DISTRIBUTION
MEAN AND VARIANCE
Example 11: 5 volunteers are needed for this problem. Each
volunteer will toss a fair coin 4 times and record the
number of heads.
Example 12: Find mean and (population) variance of 5
(random) numbers generated in Example 11 and verify if
the mean and variance are nearly same as E(X) and V(X)
computed in Example 10.
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READING AND EXERCISES
• Reading: pp. 225-234
• Exercises: 6.68, 6.70a, 6.74, 6.78
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