A binomial experiment is defined as an experiment that is repeated a fixed number of times, n, with only two possible outcomes - success or failure. The probability of success, p, is constant across all trials. Common notation includes n for the number of trials, p for the probability of success, q for the probability of failure (where q = 1 - p), and x for the number of successful trials out of n. An example is rolling a pair of dice 5 times to see how many doubles (pairs where both dice show the same number) occur.

1. Definition Binomial Experiment:

2. Definition Binomial Experiment:
►experiment is repeated for fixed number
of trials

3. Definition Binomial Experiment:
►experiment is repeated for fixed number
of trials
► two possible outcomes:

4. Definition Binomial Experiment:
►experiment is repeated for fixed number
of trials
► two possible outcomes:
► Success or Failure

5. Definition Binomial Experiment:
►experiment is repeated for fixed number
of trials
► two possible outcomes:
► Success or Failure
► probability of success is constant

6. Definition Binomial Experiment:
►experiment is repeated for fixed number
of trials
► two possible outcomes:
► Success or Failure
► probability of success is constant
► x counts the number of successful trials

7. Notation
Symbol Description
n the number of times a trial is
repeated

8. Notation
Symbol Description
n the number of times a trial is
repeated
p the probability of success

9. Notation
Symbol Description
n the number of times a trial is
repeated
p the probability of success
q the probability of failure

10. Notation
Symbol Description
n the number of times a trial is
repeated
p the probability of success
q the probability of failure
x the number of successes out of n
trials

11. Example You roll a pair of 6-sided dice 5 times, and note
whether or not you rolled a pair of doubles like
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)

12. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials

13. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials

14. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials

15. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials

16. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials

17. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials
► p = 6/36, the probability of success
► A binomial experiment has a fixed rate
of success

18. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials
► p = 6/36, the probability of success
► A binomial experiment has a fixed rate
of success
► q = 30/36, the probability of failure
► q = 1-p

19. Example You roll a pair of 6-sided dice 5 times, and note whether or
not you rolled a pair of doubles like (1, 1), (2, 2), (3, 3), (4, 4),
(5, 5), (6, 6)
► n =5 (you rolled the dice 5 times)
► A binomial experiment has a fixed
number of trials
► p = 6/36, the probability of success
► A binomial experiment has a fixed rate
of success
► q = 30/36, the probability of failure
► q = 1-p
► x = {0, 1, 2, 3, 4, 5}, the number of
successes out of n trials

20. Review Binomial experiments:

21. Review Binomial experiments:
• fixed number of trials, n

22. Review Binomial experiments:
• fixed number of trials, n
• Probability of success, p, is the same
for all trials

23. Review Binomial experiments:
• fixed number of trials, n
• Probability of success, p, is the same
for all trials
• probability of failure, q, is the
compliment of p (q = 1 – p).

24. Review Binomial experiments:
• fixed number of trials, n
• Probability of success, p, is the same
for all trials
• probability of failure, q, is the
compliment of p (q = 1 – p).
• x is the number of successes out of n
trials, x = 0, 1, 2, 3, …, n