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Copyright © 1998, Triola, Elementary Statistics
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Binomial ExperimentsBinomial ExperimentsSection 4-3 & Section 4-4Section 4-3 & Section 4-4
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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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
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DefinitionsBinomial 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.
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Notation for Binomial Distributions
S represents ‘success’
F represents ‘failure’
n = fixed number of trialsx = specific number of successes
p = probability of success in one trial q = probability of failure in one trial
P(x) = probability of getting exactly x success among n trials
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Binomial Probability Formula
Method 1
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Binomial Probability Formula
P(x) = • px • qn–xn ! (n – x )! x!
Method 1
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Binomial Probability Formula
P(x) = • px • qn–xn ! (n – x )! x!
Method 1
P(x) = nCx • px • qn–x
for calculators with nCr key, where r = x
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Table A-1 in Appendix A
Method 2
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Binomial Probability Distribution for n = 15 and p = 0.10
n
15 0. . .1. . .2. . .3. . .4. . .5. . .6. . .7. . .8. . .9. . .
10. . .11. . .12. . .13. . .14. . .15. . .
x
p
0.10
2063432671290430100020+0+0+0+0+0+0+0+0+
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Binomial Probability Distribution for n = 15 and p = 0.10
n
15 0. . .1. . .2. . .3. . .4. . .5. . .6. . .7. . .8. . .9. . .
10. . .11. . .12. . .13. . .14. . .15. . .
x
p
0.10
2063432671290430100020+0+0+0+0+0+0+0+0+
x P(x)
0123456789
101112131415
0.2060.3430.2670.1290.0430.0100.002
0+0+0+0+0+0+0+0+0+
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Use Computer Software or the TI-83 Calculator
STATDISK
Minitab
TI-83
Method 3
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P(x) = • px • qn–xn ! (n – x )! x!
Probability forone arrangement
Binomial Probability Formula
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P(x) = • px • qn–xn ! (n – x )! x!
Number of arrangements
Probability forone arrangement
Binomial Probability Formula
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For Any Probability Distribution:
Formula 4-1 µ = x • P(x)
Formula 4-3 2= [x 2 • P(x) ] – µ 2
Recall:Recall:
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For Any Probability Distribution:
Formula 4-1 µ = x • P(x)
Formula 4-3 2= [x 2 • P(x) ] – µ 2
Formula 4-4 = [x 2 • P(x) ] – µ 2
Recall:Recall:
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For a Binomial Distribution:
• Formula 4-7 µ = n • p
• Formula 4-8 2= n • p • q
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For a Binomial Distribution:
• Formula 4-7 µ = n • p
• Formula 4-8 2= n • p • q
Formula 4-9 = n • p • q