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

2. 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

3. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 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.

4. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Notation for Binomial Distributions S represents ‘success’ F represents ‘failure’ n = fixed number of trials x = 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

5. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Binomial Probability Formula Method 1

6. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Binomial Probability Formula  P(x) = p x q n–x n !n ! ( n – x ) ! x ! Method 1

7. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Binomial Probability Formula  P(x) = p x q n–x n !n ! ( n – x ) ! x ! Method 1  P(x) = n C x p x q n–x for calculators with n C r key, where r = x

8. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Table A-1 in Appendix A Method 2

9. 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 0.10 206 343 267 129 043 010 002 0+

10. 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 0.10 206 343 267 129 043 010 002 0+ x P( x ) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.206 0.343 0.267 0.129 0.043 0.010 0.002 0+

11. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Use Computer Software or the TI-83 Calculator  STATDISK  Minitab  TI-83 Method 3

12. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 P(x) = p x q n–x n !n ! ( n – x ) ! x ! Probability for one arrangement Binomial Probability Formula

13. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 P(x) = p x q n–x n !n ! ( n – x ) ! x ! Number of arrangements Probability for one arrangement Binomial Probability Formula

14. 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:

15. 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:

16. 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

17. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 For a Binomial Distribution: Formula 4-7 µ = n p Formula 4-8  2  = n p q Formula 4-9  = n p q