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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 1
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 2
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 3
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 4
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 5
Binomial Probability Formula
Method 1
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 6
Binomial Probability Formula
P(x) = • px • qn–xn ! (n – x )! x!
Method 1
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 7
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 8
Table A-1 in Appendix A
Method 2
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 9
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+
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 10
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+
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 11
Use Computer Software or the TI-83 Calculator
STATDISK
Minitab
TI-83
Method 3
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 12
P(x) = • px • qn–xn ! (n – x )! x!
Probability forone arrangement
Binomial Probability Formula
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 13
P(x) = • px • qn–xn ! (n – x )! x!
Number of arrangements
Probability forone arrangement
Binomial Probability Formula
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 14
For Any Probability Distribution:
Formula 4-1 µ = x • P(x)
Formula 4-3 2= [x 2 • P(x) ] – µ 2
Recall:Recall:
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 15
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:
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 16
For a Binomial Distribution:
• Formula 4-7 µ = n • p
• Formula 4-8 2= n • p • q