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Binomial Probability

Binomial Probability. Features of a Binomial Experiment. 1. There are a fixed number of trials. We denote this number by the letter n. Features of a Binomial Experiment. 2. The n trials are independent and repeated under identical conditions. Features of a Binomial Experiment.

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Binomial Probability

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  1. Binomial Probability

  2. Features of a Binomial Experiment 1. There are a fixed number of trials. We denote this number by the letter n.

  3. Features of a Binomial Experiment 2. The n trials are independent and repeated under identical conditions.

  4. Features of a Binomial Experiment 3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.

  5. Features of a Binomial Experiment 4. For each individual trial, the probability of success is the same. We denote the probability of success by p and the probability of failure by q. Since each trial results in either success or failure, p + q = 1 and q = 1 – p.

  6. Features of a Binomial Experiment 5. The central problem is to find the probability of r successes out of n trials.

  7. Binomial Experiments • Repeated, independent trials • Number of trials =n • Two outcomes per trial: success (S) and failure (F) • Number of successes = r • Probability of success = p • Probability of failure = q = 1 – p

  8. A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times.Is this a binomial experiment?

  9. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = failure =

  10. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = hitting the target failure = not hitting the target

  11. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = Probability of failure =

  12. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = 0.70 Probability of failure = 1 – 0.70 = 0.30

  13. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _____ trials.

  14. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _8__ trials.

  15. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times.We wish to compute the probability of six successes out of eight trials. In this case r = _____.

  16. Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _ 6__.

  17. Binomial Probability Formula

  18. Calculating Binomial Probability Given n = 6, p = 0.1, find P(4):

  19. Calculating Binomial Probability A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. n = 8, p = 0.7, find P(6):

  20. Table for Binomial Probability Table 3 Appendix II

  21. Using the Binomial Probability Table • Find the section labeled with your value of n. • Find the entry in the column headed with your value of p and row labeled with the r value of interest.

  22. Using the Binomial Probability Table • n = 8, p = 0.7, find P(6):

  23. Find the Binomial Probability Suppose that the probability that a certain treatment cures a patient is 0.30. Twelve randomly selected patients are given the treatment. Find the probability that: a. exactly 4 are cured. b. all twelve are cured. c. none are cured. d. at least six are cured.

  24. Exactly four are cured: n = r = p = q =

  25. Exactly four are cured: n = 12 r = 4 p = 0.3 q = 0.7 P(4) = 0.231

  26. All are cured: n = 12 r = 12 p = 0.3 q = 0.7 P(12) = 0.000

  27. None are cured: n = 12 r = 0 p = 0.3 q = 0.7 P(0) = 0.014

  28. At least six are cured: r = ?

  29. At least six are cured: r = 6, 7, 8, 9, 10, 11, or 12 P(10) = .000 P(11) = .000 P(12) = .000 P(6) = .079 P(7) = .029 P(8) = .008 P(9) = .001

  30. At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12) = .079 + .029 + .008 + .001 + .000 + .000 + .000 = .117

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