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Binomial Experiments
Section 4-3 & Section 4-4
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Copyright © 1998, Triola, Elementary Statistics
Addison
Wesley
Longman
Addison
Wesley Longman
1
Example
 Experiment
Flip a coin 10 times.
Let
x = # of times that the coin lands on its head
Then we call
the experiment a binomial experiment
x is called a binomial random variable
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
2
Definitions
Binomial Experiment
1. The experiment must have a fixed number of trials.
2. The trials must be independent. (The outcome of
any individual trial doesn’t affect the probabilities
in the other trials.)
3. Each trial must have all outcomes classified into
two categories.
4. The probabilities must remain constant for each
trial.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Notation for Binomial
Distributions
S represents ‘success’
F represents ‘failure’
n = fixed number of trials
x =
p=
q=
P(x) =
specific number of successes
probability of success in one trial
probability of failure in one trial
probability of getting exactly x
success among n trials
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
4
Method 1
Binomial Probability
Formula
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Method 1
Binomial Probability
Formula
 P(x)
n!
= (n – x )! x! •
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
px
•
qn–x
6
Method 1
Binomial Probability
Formula
 P(x) =
n!
•
(n – x )! x!
 P(x) = nCx • px
•
px
•
qn–x
qn–x
for calculators with nCr key, where r = x
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Method 2
Table A-1 in Appendix A
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Addison Wesley Longman
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Binomial Probability Distribution
for n = 15 and p = 0.10
p
n
x
15
0. . .
1. . .
2. . .
3. . .
4. . .
5. . .
6. . .
7. . .
8. . .
9. . .
10. . .
11. . .
12. . .
13. . .
14. . .
15. . .
0.10
206
343
267
129
043
010
002
0+
0+
0+
0+
0+
0+
0+
0+
0+
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
9
Binomial Probability Distribution
for n = 15 and p = 0.10
p
n
x
15
0. . .
1. . .
2. . .
3. . .
4. . .
5. . .
6. . .
7. . .
8. . .
9. . .
10. . .
11. . .
12. . .
13. . .
14. . .
15. . .
0.10
x
206
343
267
129
043
010
002
0+
0+
0+
0+
0+
0+
0+
0+
0+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
P(x)
0.206
0.343
0.267
0.129
0.043
0.010
0.002
0+
0+
0+
0+
0+
0+
0+
0+
0+
10
Method 3
Use Computer Software
or the TI-83 Calculator
 STATDISK
 Minitab
 TI-83
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Binomial Probability
Formula
P(x)
n!
= (n – x )! x! •
px
•
qn–x
Probability for
one arrangement
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Addison Wesley Longman
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Binomial Probability
Formula
P(x)
n!
= (n – x )! x! •
Number of
arrangements
px
•
qn–x
Probability for
one arrangement
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Recall: For
Any Probability
Distribution:
= Sx • P(x)
Formula 4-1 µ
Formula 4-3 s
2
= [Sx • P(x) ] – µ
2
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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14
Recall: For
Any Probability
Distribution:
= Sx • P(x)
Formula 4-1 µ
Formula 4-3 s
Formula 4-4 s
2
= [Sx • P(x) ] – µ
2
=
2
[Sx • P(x) ] – µ
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Addison Wesley Longman
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2
15
For a Binomial Distribution:
• Formula 4-7
• Formula 4-8
µ =n•p
s =n•p•q
2
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Addison Wesley Longman
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For a Binomial Distribution:
• Formula 4-7
• Formula 4-8
µ =n•p
s =n•p•q
2
Formula 4-9 s =
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
n•p•q
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