1. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. CHAPTER 6 Discrete Probability Distributions

2. Chap 22

3. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Discrete Random Variables 6.1

4. 6-4 A random variable is a numerical outcome from a probability experiment. Its value is determined by chance, aka dumb luck. Random variables are denoted using letters such as “x, y, and z”.

5. 6-5 A discrete random variable has countable number of values. Its values can be plotted on a number line in an uninterrupted fashion, i.e., only the values from 0, 1, 2, 3, and 4. Think “digital” watch

6. 6-6 A continuous random variable has infinitely many values. Its values can be measured, not counted. Its values can be plotted on a number line in an uninterrupted fashion, i.e., all the values from 0 thru 4. Think “analog” or sweep-second hand watch

7. 6-7 (a)The number of light bulbs that burn out in a room with 10 light bulbs in the next year. (b) The number of leaves on a randomly selected oak tree. (c) The length of time between calls to 911. EXAMPLEDistinguishing Between Discrete and Continuous Random Variables Discrete; x = 0, 1, 2, … Continuous; t > 0

8. 6-8 A probability distribution provides all possible values of the outcome random variable “x” and the corresponding probabilities of each outcome occurring. A probability distribution can be in the form of a table, graph or math formula.

9. 6-9 “x” represents the survey results of the number of movies streamed on Netflix each month at random homes in the county. P(x), read “P of x”, is the probability of each possible outcome occurring. So, if you called a random home, the prob they would say they streamed 3 movies is 0.10… xP(x)P(x) 00.06 10.58 20.22 30.10 40.03 50.01 EXAMPLEA Discrete Probability Distribution

10. Rules for a Discrete Probability Distribution Σ P(x) = 1: The sum of all Probs = 1 0 ≤ P(x) ≤ 1 : Each individual Prob is between 0 and 1 Rules for a Discrete Probability Distribution Σ P(x) = 1: The sum of all Probs = 1 0 ≤ P(x) ≤ 1 : Each individual Prob is between 0 and 1 5-10

11. 6-11 A probability histogram is a graph in which the horizontal axis plots the values of “x”, and the vertical axis plots the probability of that value of “x” occurring as an outcome.

12. 6-12 Probability histogram and probability distribution representing the number of movies streamed on Netflix each month. EXAMPLEDrawing a Probability Histogram xP(x)P(x) 00.06 10.58 20.22 30.10 40.03 50.01

13. Mean of a Discrete Random Variable The mean of a discrete random variable is given by the formula where x is the value of the variable and P(x) is the probability of observing that value x. Mean of a Discrete Random Variable The mean of a discrete random variable is given by the formula where x is the value of the variable and P(x) is the probability of observing that value x. 5-13

14. 6-14 Compute the mean of this distribution, which represents the mean number of DVDs a random customer rents from a video store during a visit. EXAMPLEComputing the Mean of a Discrete Random Variable xP(x)P(x) 00.06 10.58 20.22 30.10 40.03 50.01

15. Interpretation of the Mean of a Discrete Random Variable Suppose an experiment (observing DVD rentals) is repeated a large number of times (Law Large Numbers). As the number of repetitions of the experiment increases, the mean value of the trials will approach μ X =1.49 DVDs rented each time. So, the first ten customers may rent 5 DVDs each, but after 500 customers have come in, their mean number of DVDs will approach 1.49 each. That is what Probability tells us ~ the expected outcome over time, not instantaneous results. Interpretation of the Mean of a Discrete Random Variable Suppose an experiment (observing DVD rentals) is repeated a large number of times (Law Large Numbers). As the number of repetitions of the experiment increases, the mean value of the trials will approach μ X =1.49 DVDs rented each time. So, the first ten customers may rent 5 DVDs each, but after 500 customers have come in, their mean number of DVDs will approach 1.49 each. That is what Probability tells us ~ the expected outcome over time, not instantaneous results. 5-15

16. 6-16 The following data represent the actual number of DVDs rented by 100 customers in a single visit. Compute the mean number of DVDs rented.

17. 6-17 As the number of trials of the DVD experiment increases, the mean number of rentals approaches the mean of the probability distribution.

18. 6-18 Because the mean of a discrete random variable represents what we would expect to happen in the long run, it is also called the expected value, E(x), of the random variable.

19. 6-19 EXAMPLEComputing the Expected Value of a Discrete Random Variable A life insurance policy will pay a discrete sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a company sells a one-year $250,000 life insurance policy to a 49-year-old woman for $530. The probability she will survive the year is 0.99791. Compute the expected value (distribution mean) of this policy to the insurance company. X ($)P(x)P(x) +5300.99791 530 – 250,000 = -249,470 0.00209 She survives She does not survive E(X) = 530(0.99791) + (-249,470)(0.00209) What does that EV represent to the company if they sell 100,000 policies – profit or loss? = $7.50

20. Larson & Farber, Elementary Statistics: Picturing the World, 3e 20 Expected Value Gain, x P (x) 9 At a raffle, 500 tickets are sold for $1 each for two prizes of $100 and $50. If you buy one ticket, what is the expected value of your gain? Because the expected value is negative, you can expect to lose an average of $0.70 for each $1 ticket you buy. Winning no prize +$99 +$4 –$1 E(x) = Σ[x P (x) ]

21. Chap 4 21 TI-84 To find the Mean (EV) of a Probability Distribution Go to Stat:Edit – enter the “x” values in List 1 and the “Prob” values in List 2 Go to Stat:Calc:1Var-Stats Enter: L1,L2 This will give μ, or the Expected Value of the outcome. This will give μ, or the Expected Value of the outcome.

22. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section The Binomial Probability Distribution 6.2

23. Criteria for a Binomial Probability Experiment A binomial experiment is: 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are only two disjoint outcomes as: Success/Failure; Pass/Fail; True/False; Yes/No, etc 4. The probability of success “p” is fixed for each trial of the experiment. Criteria for a Binomial Probability Experiment A binomial experiment is: 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are only two disjoint outcomes as: Success/Failure; Pass/Fail; True/False; Yes/No, etc 4. The probability of success “p” is fixed for each trial of the experiment. 5-23

24. Binomial Probability Distribution There are “n” independent trials of the experiment. Let “p” denote the probability of success so that “q” = (1 – p) is the probability of failure. Let “x” be a binomial discrete variable that denotes the number of successes in “n” trials So…. 0 < x < n. Binomial Probability Distribution There are “n” independent trials of the experiment. Let “p” denote the probability of success so that “q” = (1 – p) is the probability of failure. Let “x” be a binomial discrete variable that denotes the number of successes in “n” trials So…. 0 < x < n. 5-24

25. 6-25 According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time arrival percentage of 79.0% in May, 2008. Suppose that 4 flights are randomly selected from May, 2008 and the number of on-time flights “x” is recorded. Construct a probability distribution for the random variable “x” using a tree diagram. EXAMPLEConstructing a Binomial Probability Distribution

26. The probability of obtaining x successes in n independent trials of a binomial experiment is given by: where “p” is the probability of success for each trial. The probability of obtaining x successes in n independent trials of a binomial experiment is given by: where “p” is the probability of success for each trial. 5-26

27. 6-27 PhraseMath Symbol “at least” or “no less than” ≥ “greater than or equal to” “more than” “greater than” > “fewer than” “less than”< “no more than” “at most” ≤ “less than or equal to “exactly” “equals” “is”= “does not equal” “is not” ≠

28. 6-28 EXAMPLEUsing the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a random sample of 20 car-owning households, what is the probability that exactly 5 have three or more cars?

29. Chap 4 29 TI-83/84 DISTR:0 is binompdf for “prob density function”. It gives you individual discrete probabilities. DISTR:0 is binompdf for “prob density function”. It gives you individual discrete probabilities. DISTR:A is binomcdf for “cum density function”. It gives you cumulative (starting from zero) discrete probabilities. DISTR:A is binomcdf for “cum density function”. It gives you cumulative (starting from zero) discrete probabilities.

30. Chap 4 30 binompdf… (n, p, x) If you wanted to know the prob that exactly 5 households have three or more cars, If you wanted to know the prob that exactly 5 households have three or more cars, … and the prob of “success” on each trial was 0.350, then … … and the prob of “success” on each trial was 0.350, then … Binompdf (20,0.350,5)= 0.1272 Binompdf (20,0.350,5)= 0.1272

31. 6-31 EXAMPLEUsing the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a random sample of 20 car-owning households, what is the probability that less than 4 have three or more cars?

32. Chap 4 32 binomcdf (n,p,x) This function accumulates prob starting with x=0 This function accumulates prob starting with x=0 For P(x = 3 or less) For P(x = 3 or less) binomcdf (20,0.35,3)=0.0444 binomcdf (20,0.35,3)=0.0444 So, the prob of having 4 or more cars is: So, the prob of having 4 or more cars is: 1 - 0.0444 = 0.9556 1 - 0.0444 = 0.9556

33. Mean (Expected Value) and Standard Deviation of a Binomial Random Variable A binomial experiment with “n” independent trials and probability of success “p” has a mean / standard deviation given by: Mean (Expected Value) and Standard Deviation of a Binomial Random Variable A binomial experiment with “n” independent trials and probability of success “p” has a mean / standard deviation given by: 5-33

34. 6-34 According to Experian Automotive, 35% of all car-owning households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard deviation of car-owning households that will have three or more cars. EXAMPLEFinding the Mean and Standard Deviation of a Binomial Random Variable So, if the data were normally distributed, 68% of these households would own between 130.46 and 149.54 cars

35. 6-35 Construct a binomial probability histograms with n = 8 and varying “p” In other words, you flip a coin 8 times. Success is (approx) : a) getting a 1 p ≈ 0.15 b) getting a 3 or less p ≈ 0.50 c) getting a 2 or more p ≈ 0.85 For each histogram, comment on the shape of the distribution. EXAMPLEConstructing Binomial Probability Histograms

36. 6-36 In other words, in 8 coin flips, you got 2 successes (rolling a 1) about 24% of the time.

37. 6-37 In other words, in 8 coin flips, you got 5 successes (rolling 3 or less) about 22% of the time.

38. 6-38 In other words, in 8 coin flips, you got 7 successes (rolling 2 or more) almost 40% of the time. μ 7/8 = 0.875

39. 6-39 μ 21/25 = 0.840

40. 6-40 μ 58/70 = 0.829

41. For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable “x” becomes bell-shaped. As a general rule, if σ > 3 then the binomial probability distribution will be approximately bell-shaped. (called a “normal” distribution). For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable “x” becomes bell-shaped. As a general rule, if σ > 3 then the binomial probability distribution will be approximately bell-shaped. (called a “normal” distribution). 5-41

42. Use The Empirical Rule (for normal distributions) to identify “unusual” observations in an experiment. 95% of the observations lie within ± 2σ from the μ Any observation that lies outside this interval may be considered “unusual” because it occurs less than 5% of the time. Use The Empirical Rule (for normal distributions) to identify “unusual” observations in an experiment. 95% of the observations lie within ± 2σ from the μ Any observation that lies outside this interval may be considered “unusual” because it occurs less than 5% of the time. 5-42

43. Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section The Poisson Probability Distribution 6.3

44. Larson & Farber, Elementary Statistics: Picturing the World, 3e 44 Poisson Distribution The Poisson (Fr: b 1781) distribution satisfies the following conditions. 1. The experiment consists of counting the number of times an event, x, occurs in a given interval. The interval can be an interval of time, area, or volume. 2. The probability of the event occurring is the same for each interval. 3.The number of occurrences in one interval is independent of the number of occurrences in other intervals. The probability of exactly x occurrences in an interval is where e  2.71818 and μ is the mean number of occurrences.

45. Larson & Farber, Elementary Statistics: Picturing the World, 3e 45 Poisson Distribution The mean number of power outages in the city of Gulf Breeze is 4 per week. Find the probability that in any given week…. a.) there are exactly 3 outages b.) there are more than 3 outages. Note: λ = μ in the Poisson formula

46. 46 Poisson Distribution a) 3 people are killed by sharks this year b) Two or three people are killed by sharks this year P(3) = 0.0076 P(2 or 3) = 0.0023 + 0.0076 = 0.0099 Statistics show that, on average, sharks kill 10 ( λ) people each year worldwide. Find the probability that….

47. Chap 4 47 DISTR:B:poissonpdf ( λ,x) The mean number of major hurricanes that strike Florida in any given year is 0.7 (7 storms every 10 years) The mean number of major hurricanes that strike Florida in any given year is 0.7 (7 storms every 10 years) What is the probability that 2 major hurricanes will strike Florida this year? What is the probability that 2 major hurricanes will strike Florida this year? poissonpdf(0.7,2) = 0.1217 poissonpdf(0.7,2) = 0.1217

48. Chap 4 48 TI-83/84 You can also use CUMULATIVE functions for Poisson distributions. This is similar to Binomcdf, where the calculator will count up all probabilities up to, and including the count that you specify. Ex: you want to calc the prob that 2 or less major hurricanes will strike Florida this year. DISTR:D poissoncdf (0.7,2) = 0.9659 (not 0.1217)

49. 49Chap 4 The following Sections on Geometric and Hypergeometric Probabilities are Optional They are for your academic interest. They will not be covered on exams.

50. 50Chap 4 Geometric Distributions

51. Larson & Farber, Elementary Statistics: Picturing the World, 3e 51 Geometric Distribution A geometric distribution satisfies the following conditions. : 1. A trial is repeated until a success occurs. 2. The repeated trials are independent of each other. 3.The probability of a success p is constant for each trial. The probability that the first success will occur on trial x is P (x) = p (q) x – 1 where q = 1 – p.

52. 52 Geometric Distribution A study has found that the probability that a person who enters a store will actually make a purchase is 0.30 Therefore, the probability that the first purchase will be made by customer number 4 is: P(4) = (.30)(.70) 3 = 0.1029

53. Chap 4 53 DISTR:D:geometpdf… (p,x) You own a clothing store. Suppose the prob you will make a sale to any given customer is 0.30 To find the prob that your first sale occurs with the 4 th customer… To find the prob that your first sale occurs with the 4 th customer… (F,F,F,S) = (0.70^3)(0.30^1) geometpdf(0.30,4) = 0.1029 geometpdf(0.30,4) = 0.1029

54. 54Chap 4 Hypergeometric Distributions

55. Hypergeometric Probability Distribution The probability of obtaining “x” successes from a sample size “n” is: where “k” is the number of successes in the population size “N”. Hypergeometric Probability Distribution The probability of obtaining “x” successes from a sample size “n” is: where “k” is the number of successes in the population size “N”. 5-55

56. 6-56 EXAMPLEA Hypergeometric Probability Distribution The Dow Jones Industrial Average (DJIA) is a collection of 30 traded companies on the US Stock Market that are meant to be representative of the entire United States economy. In one certain month 18 of the 30 stocks (60%) in the DJIA increased in value. What is the probability that an investor who randomly invests in 4 of these 30 stocks at the beginning of that month found that 3 of his 4 stocks increased in value?

57. 6-57 EXAMPLEA Hypergeometric Probability Distribution k = 18 N = 30 and x = 3 n = 4

58. 6-58 EXAMPLEA Hypergeometric Probability Distribution Try this Hypergeometric problem: Women make up 54% of the adult population of the US. What is the probability that, if you selected 17 US adults, that exactly 13 of them would be women? Ans: 0.0325 or: if you picked 17 adults from the population 10,000 times, 325 of those times there would be 13 women out of the 17.

59. 59 Jury Selection Problem There are 60 women and 40 men waiting outside the courtroom in the jury pool. There are 60 women and 40 men waiting outside the courtroom in the jury pool. Assuming the probability of being selected for the jury is the same for each person, what is the probability that the 12 person jury selected will consist of 7 men and 5 women? Assuming the probability of being selected for the jury is the same for each person, what is the probability that the 12 person jury selected will consist of 7 men and 5 women?

60. 60 End Chap 6 ! Are you having fun yet?? Are you having fun yet?? Don’t worry, it gets even better! Don’t worry, it gets even better!

61. Chap 261