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Chapter Six

Chapter Six. The Binomial Probability Distribution and Related Topics. Statistical Experiment or Observation. any process by which measurements are obtained. Examples of Statistical Experiments. Counting the number of books in the College Library

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Chapter Six

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  1. Chapter Six The Binomial Probability Distribution and Related Topics

  2. Statistical Experiment or Observation • any process by which measurements are obtained

  3. Examples of Statistical Experiments • Counting the number of books in the College Library • Counting the number of mistakes on a page of text • Measuring the amount of rainfall in your state during the month of June

  4. Random Variable • a quantitative variable that assumes a value determined by chance or random outcome

  5. Discrete Random Variable A discrete random variable is a quantitative random variable that can take on only a finite number of values or a countable number of values. Example: the number of books in the College Library

  6. Continuous Random Variable A continuous random variable is a quantitative random variable that can take on any of the countless number of values in a line interval. Example: the amount of rainfall in your state during the month of June

  7. Probability Distribution • an assignment of probabilities to the specific values of the random variable or to a range of values of the random variable

  8. Probability Distribution of a Discrete Random Variable • A probability is assigned to each value of the random variable. • The sum of these probabilities must be 1.

  9. Probability distribution for the rolling of an ordinary die xP(x) 1 2 3 4 5 6

  10. Frequency Distribution andProbability Distribution

  11. A Probability Histogram

  12. Mean and standard deviation of a discrete probability distribution • Mean = m = expectation or expected value, • the long-run average Formula:m = S x P(x)

  13. Standard Deviation

  14. Finding the mean: xP(x) xP(x) 0 .3 1 .3 2 .2 3 .1 4 .1 0 .3 .4 .3 .4 m = å x P(x) = 1.4 1.4

  15. Finding the standard deviation x P(x) x – m ( x – m) 2 ( x – m) 2 P(x) 0 .3 1 .3 2 .2 3 .1 4 .1 .588 .048 .072 .256 .676 – 1.4 – 0.4 .6 1.6 2.6 1.96 0.16 0.36 2.56 6.76 1.64

  16. Standard Deviation 1.28

  17. Binomial Probabilities • Probabilities associated with binomial or Bernoulli experiments where there are two possible outcomes for each trial

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

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

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

  21. 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.

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

  23. 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

  24. A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times.Is this a binomial experiment?

  25. Check the Features of a Binomial Experiment • 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

  26. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. success = failure =

  27. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. success = hitting the target failure = not hitting the target

  28. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. Probability of success = Probability of failure =

  29. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. Probability of success = 0.70 Probability of failure = 1 – 0.70 = 0.30

  30. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. In this experiment there are n = trials.

  31. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. In this experiment there are n = 8 trials.

  32. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = .

  33. Binomial Experiment A marksman takes eight shots at a target. He normally hits the target 70% of the time. Find the probability that he hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = 6 .

  34. Binomial Probability Formula

  35. Binomial Probability Formula n = number of trials p = probability of success on each trial q = 1  p = probability of failure on each trial r = number of successes

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

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

  38. Table for Binomial Probability • Appendix • Table 2 • Pages A2 – A5

  39. A Section of the Binomial Probability Table

  40. 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.

  41. Using the Binomial Probability Table n = 8, p = 0.25, find P(6):

  42. Using the Binomial Probability Table n = 8, p = 0.25, find P(6):

  43. Using the Binomial Probability Table n = 8,p = 0.25, find P(6):

  44. Using the Binomial Probability Table n = 8,p = 0.25, find P(6):

  45. Using the Binomial Probability Table For n = 8, p = 0.25, P(6) = 0.004

  46. 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.

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

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

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

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

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