Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Mistah Flynn

1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Mistah Flynn BHS Chapter Six Probability Distributions and Binomial Distributions

2. Statistical Experiment A statistical experiment or observation is any process by which an 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 • Counting the number of siblings you have • “Start of the Year” data????

4. Random Variable a quantitative variable that assumes a value determined by chance

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. Discrete Random Variables Continuous Discrete Question about… Random Variable Let x = Type? Family x = Number of dependents in size family reported on tax return Distance from x = Distance in miles from home to store home to the store site Own dog x = 1 if own no pet; or cat = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

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

9. 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.

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

11. Features of a Probability Distribution x P(x) 1 2 3 4 5 6 Probabilities must be between zero and one (inclusive)  P(x) =1

12. Probability Histogram P(x) | | | | | | | 1 2 3 4 5 6

13. Mean and standard deviation of a discrete probability distribution Mean =  = expectation or expected value, the long-run average Formula:  =  x P(x)

14. Standard Deviation

15. Finding the mean: x P(x) x P(x) 0 .3 1 .3 2 .2 3 .1 4 .1 0 .3 .4 .3 .4  =  x P(x) = 1.4 1.4

16. Finding the standard deviation x P(x) x –  ( x – ) 2 ( x – ) 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

17. Standard Deviation 1.28

18. Features of a Probability Distribution x P(x) 1 2 3 4 5 6 Probabilities must be between zero and one (inclusive)  P(x) =1

19. Finding the mean a die roll: x P(x) x P(x) 1 .167 2 .167 3 .167 4 .167 5 .167 6 .167 .167 .333 .5 .667 .833 1.000  =  x P(x) = 3.5 3.500

20. Finding the standard deviation x P(x) x –  ( x – ) 2 ( x – ) 2 P(x) 1 .167 2 .167 3 .167 4 .167 5 .167 6 .167 1.042 .375 .042 .042 .375 1.042 6.25 2.25 0.25 0.25 2.25 6.25 -2.5 -1.5 -0.5 0.5 1.5 2.5 2.918

21. Standard Deviationof a single die roll: 1.707

22. Expected value and standard deviation… At a carnival, you decide to play spinner the wheel for a prize for $2.00. If you land in the red area, you win $4. If you land in the white area, you win $1. If you land in the blue area, you win nothing. What are your expected earnings when you play the game once? Ten times? Formula:  =  x P(x)

23. Standard Deviation

24. Finding the mean: x P(x) x P(x) red .333 white .333 blue .333 ???? ??? ??  =  x P(x) = ?

25. Finding the mean: x P(x) x P(x) $2 .333 -$1 .333 -$2 .333 $0.67 -$0.33 -$0.67  =  x P(x) = -$0.33 If you do this ten times, you should expect to lose -$3.33

26. Finding the standard deviation x P(x) x –  ( x – ) 2 ( x – ) 2 P(x) $2 .333 -$1 .333 -$2 .333 $1.81 $0.15 $0.93 $2.33 –$0.67 –$1.67 $5.43 $0.45 $2.79 $2.89

27. Standard Deviation $1.70

28. Binomial Probability A special kind of discrete probability distribution with only 2 random variables

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

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

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

32. 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.

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

34. 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

35. 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?

36. 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 =

37. 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

38. 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 =

39. 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

40. 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.

41. 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.

42. 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 = _____.

43. 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__.

44. Binomial Probability as a Counting Method (FCP) = ? x ? x ? x ? x ? x ? x ? x ?  if taking 8 shots, we would multiply the probabilities together… = 0.7 x 0.7 x 0.7 x 0.7 x 0.7 x 0.7 x 0.3 x 0.3  if taking 8 shots, and 6 were successful and two were not… = 0.0106 … but that is only if the first 6 were successful and the last two were not… = 0.7 x 0.3 x 0.7 x 0.7 x 0.3 x 0.7 x 0.7 x 0.7  if taking 8 shots, and 6 were successful and two were not…in any particular order…how many different ways could it happen?

45. What are Binomial Experiments? • The have a fixed number of _________ which we call the variable _____. • The only outcomes for the binomial experiment are ___________ and ____________. • Each trial in the experiment is ____________ and repeatable. • Number of successes is the variable ____. • Probability of success is the variable ____. • Essentially each binomial experiment tries to the find the ____________ of __________ out of a given number of ____________.

46. Binomial Probability Formula

47. 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):

48. Table for Binomial Probability Table 3 Appendix II

49. 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.

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