Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Binomial Experiments...

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



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



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.



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



Binomial Probability Formula

Method 1




P(x) = • px • qn–xn ! (n – x )! x!

Method 1




P(x) = • px • qn–xn ! (n – x )! x!

Method 1

P(x) = nCx • px • qn–x

for calculators with nCr key, where r = x



Table A-1 in Appendix A

Method 2



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+



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+



Use Computer Software or the TI-83 Calculator

STATDISK

Minitab

TI-83

Method 3



P(x) = • px • qn–xn ! (n – x )! x!

Probability forone arrangement




P(x) = • px • qn–xn ! (n – x )! x!

Number of arrangements

Probability forone arrangement




For Any Probability Distribution:

Formula 4-1 µ = x • P(x)

Formula 4-3 2= [x 2 • P(x) ] – µ 2

Recall:Recall:



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:



For a Binomial Distribution:

• Formula 4-7 µ = n • p

• Formula 4-8 2= n • p • q



For a Binomial Distribution:

• Formula 4-7 µ = n • p

• Formula 4-8 2= n • p • q

Formula 4-9 = n • p • q

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Binomial Experiments...

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