1. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-1 Business Statistics: A Decision-Making Approach 8 th Edition Chapter 5 Discrete Probability Distributions

2. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-2 Chapter Goals After completing this chapter, you should be able to: Calculate and interpret the expected value of a discrete probability distribution Apply the binomial distribution to business problems Compute probabilities for the Poisson and hypergeometric distributions Recognize when to apply discrete probability distributions to decision making situations

3. Random variable vs. Probability distribution When the value of a variable is the outcome of a statistical experiment, that variable is a random variable. Let the variable X represent the number of Heads that result from the previous experiment. The variable X can take on the values 0, 1, or 2. In this previous example, X is a random variable A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. Just like the previous table Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-3

4. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-4 Experiment: Toss 2 Coins. Let x = # heads. T T Random variable vs. Probability distribution 4 possible outcomes T T H H HH 0 1 2 x x Value Probability 0 1/4 = 0.25 1 2/4 = 0.50 2 1/4 = 0.25 0.50 0.25 Probability

5. Cumulative Probability and Cumulative Probability Distribution A cumulative probability refers to the probability that the value of a random variable falls within a specified range. A cumulative probability distribution can be represented by a table or an equation. See the next slide Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-5

6. Cumulative Probability Distribution Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-6 Number of heads: xProbability: P(X = x) Cumulative Probability: P(X < x) 00.25 10.500.75 20.251.00

7. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-7 A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes: number of complaints per day number of TV’s in a household number of rings before the phone is answered Only two possible outcomes: gender: male or female defective: yes or no spreads peanut butter first vs. spreads jelly first Discrete Random Variable

8. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-8 Continuous Random Variable A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

9. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-9 If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution. Discrete Probability Distribution Number of headsProbability 00.25 10.50 20.25

10. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-10 Discrete Random Variable Mean (formula) Expected Value (or mean ) of a discrete distribution (Weighted Average) E(x) =  xP(x) Example: Toss 2 coins, x = # of heads, compute expected value of x: E(x) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 x P(x) 0 0.25 1 0.50 2 0.25

11. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-11 Standard Deviation of a discrete distribution where: E(x) = Expected value of the random variable (done!) x = Values of the random variable P(x) = Probability of the random variable having the value of x Discrete Random Variable Standard Deviation (formula)

12. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-12 Example: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1) Discrete Random Variable Standard Deviation (continued) Possible number of heads = 0, 1, or 2

13. Using Excel Use Excel for calculating: Discrete Random Variable Mean Discrete Random Variable Standard Deviation Download and open “Binomial Dist. Examples” Excel file… And then, try the example on the first tap… Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-13

14. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-14 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Discrete Probability Distributions Normal Uniform Exponential Ch. 5 Ch. 6

15. Binomial Experiment The experiment involves repeated trials. Each trial has only two possible outcomes - a success or a failure (i.e., head/tail, goal/no goal). The probability that a particular outcome will occur on any given trial is constant. 0.5 every trial All of the trials in the experiment are independent. The outcome on one trial does not affect the outcome on other trials. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-15

16. Binomial Experiment Example Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-16 Outcome, x Binomial probability, P(X = x) Cumulative probability, P(X < x) 0 Heads0.125 1 Head0.3750.500 2 Heads0.3750.875 3 Heads0.1251.000

17. Binomial Probability A binomial probability refers to the probability of getting EXACTLY n successes in a specific number of trials. Example: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. Using the table on the previous slide, that probability (0.375) would be an example of a binomial probability. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-17

18. Cumulative Binomial Probability Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range. Using the table on the previous slide, the probability of getting AT MOST 2 Heads (less than equal to: <) in 3 coin tosses is an example of a cumulative probability. 0 heads (0.125) + 1 head (0.375) + 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-18

19. Notation associated with cumulative binomial probability The probability of getting FEWER THAN 2 successes is indicated by P(X < 2). The probability of getting AT MOST 2 successes is indicated by P(X < 2). The probability of getting AT LEAST 2 successes is indicated by P(X > 2). The probability of getting MORE THAN 2 successes is indicated by P(X > 2). Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-19

20. Binomial Distribution A binomial distribution is a probability distribution. It refers to the probabilities associated with the number of successes in a binomial experiment. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-20

21. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-21 n = 5 p = 0.1 n = 5 p = 0.5 Mean 0.2.4.6 012345 X P(X).2.4.6 012345 X P(X) 0 Binomial Distribution The shape of the binomial distribution depends on the values of p and n Here, n = 5 and p = 0.1 Here, n = 5 and p = 0.5 Try the “Binomial Distribution Simulation” on the class website

22. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-22 P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of successes in sample, (x = 0, 1, 2,..., n) p = probability of “success” per trial q = probability of “failure” = (1 – p) n = number of trials (sample size) P(x) n x ! nx pq x n x ! () !    Binomial Distribution Formula

23. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-23 Binomial Distribution Example Example: 35% of all voters support Proposition A. If a random sample of 10 voters is polled, what is the probability that exactly three of them support the proposition? i.e., find P(x = 3) if n = 10 and p = 0.35 : There is a 25.22% chance that 3 out of the 10 voters will support Proposition A

24. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-24 Examples: n = 10, p = 0.35, x = 3: P(x = 3|n =10, p = 0.35) = 0.2522 n = 10, p = 0.75, x = 2: P(x = 2|n =10, p = 0.75) = 0.0004 Using Binomial Tables n = 10 xp=.15p=.20p=.25p=.30p=.35p=.40p=.45p=.50 0 1 2 3 4 5 6 7 8 9 10 0.1969 0.3474 0.2759 0.1298 0.0401 0.0085 0.0012 0.0001 0.0000 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010 10 9 8 7 6 5 4 3 2 1 0 p=.85p=.80p=.75p=.70p=.65p=.60p=.55p=.50x

25. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-25 Using PHStat Select: Add-Ins / PHStat / Probability & Prob. Distributions / Binomial…

26. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-26 Using PHStat Enter desired values in dialog box Here:n = 10 p = 0.35 Output for x = 0 to x = 10 will be generated by PHStat Optional check boxes for additional output

27. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-27 P(x = 3 | n = 10, p = 0.35) = 0.2522 PHStat Output P(x > 5 | n = 10, p = 0.35) = 0.0949

28. Cumulative Binomial Probability Refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit). Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-28

29. Example 1 What is the cumulative binomial probability of obtaining 45 or fewer heads in 100 tosses of a coin? This would be the sum of all these individual binomial probabilities. b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + …….. + b(x = 45; 100, 0.5) b(x < 45; 100, 0.5) = 0.184 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-29

30. Example 2 The probability that a student is accepted to a prestigious college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted? b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3) b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087 b(x < 2; 5, 0.3) = 0.8369 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-30

31. More Binomial Distribution Please see the textbook….. Now, Try the binominal distribution using Excel… Use previously downloaded Excel file… Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-31

32. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-32 The Poisson Distribution Characteristics of the Poisson Distribution: The outcomes of interest are rare relative to the possible outcomes The average number of outcomes of interest per time or space interval is The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest The probability that an outcome of interest occurs in a given segment is the same for all segments e.g. the number of customers per hour or the number of bags lost per flight

33. Why Poisson Distribution? When the total # of possible outcomes (success + failure) cannot be determined The # of potholes that develop per mile Found the pothole means “Success”… How could it count the # of non-potholes? The # of emergencies medical service agency respond to in one hour Responded means “Success”… How could it count the # of non-response – how many calls did not receive? Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-33

34. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-34 Poisson Distribution Summary Measures Mean Variance and Standard Deviation where = number of successes in a segment of unit size t = the size of the segment of interest

35. Poisson distribution – Using Excel Excel can be used to find both the cumulative probability as well as the point estimated probability for a Poisson experiment. In order to get Excel to calculate poisson probabilities, you have to use the following syntax in a cell. =poisson (x; mean; cumulative) Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-35

36. Poisson distribution – Using Excel X is the number of events. Mean is simply the mean of the variable. Cumulative has the options of FALSE and TRUE. If you choose FALSE, Excel will return probability of only and only the x number of events happening. If you choose TRUE, Excel will return the cumulative probability of the event x or less happening. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-36

37. Example: Point Estimate A bakery has avergae 6 customers during a business hour. We then wish to calculate the probability of the event that exactly 4 customers enter the store in the next hour. That is: x = 4, mean = 6 and cumulative = FALSE Would be written in excel as: =poisson(4;6;FALSE) And return the probability of 0.133853 = 13.3853% Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-37

38. Example: Cumulative A bakery has average 6 customers during a business hour. We then wish to calculate the probability of the event that 4 customers or less enter the store in the next hour. That is: x = 4, mean = 6 and cumulative = TRUE Would be written in excel as: =poisson(4;6;TRUE) And return the probability of 0.285057 = 28.5057% Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-38

39. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-39 Graph of Poisson Probabilities X t = 0.50 0123456701234567 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 P(x = 2) = 0.0758 Graphically: =.05 and t = 100

40. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-40 Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameters and t: t = 0.50 t = 3.0

41. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-41 Poisson Distribution Formula where: t = size of the segment of interest x = number of successes in segment of interest = expected number of successes in a segment of unit size e = base of the natural logarithm system (2.71828...)

42. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-42 Using Poisson Tables X t 0.100.200.300.400.500.600.700.800.90 0123456701234567 0.9048 0.0905 0.0045 0.0002 0.0000 0.8187 0.1637 0.0164 0.0011 0.0001 0.0000 0.7408 0.2222 0.0333 0.0033 0.0003 0.0000 0.6703 0.2681 0.0536 0.0072 0.0007 0.0001 0.0000 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 0.5488 0.3293 0.0988 0.0198 0.0030 0.0004 0.0000 0.4966 0.3476 0.1217 0.0284 0.0050 0.0007 0.0001 0.0000 0.4493 0.3595 0.1438 0.0383 0.0077 0.0012 0.0002 0.0000 0.4066 0.3659 0.1647 0.0494 0.0111 0.0020 0.0003 0.0000 Example: Find P(x = 2) if = 0.05 and t = 100

43. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-43 The Hypergeometric Distribution “n” trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent The probability changes from trial to trial Concerned with finding the probability of “x” successes in the sample where there are “X” successes in the population

44. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-44 Hypergeometric Distribution Formula. Where N = population size X = number of successes in the population n = sample size x = number of successes in the sample n – x = number of failures in the sample (Two possible outcomes per trial: success or failure)

45. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-45 Hypergeometric Distribution Example ■Example: 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective? N = 10n = 3 X = 4 x = 2

46. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-46 Hypergeometric Distribution in PHStat Select: Add-Ins / PHStat / Probability & Prob. Distributions / Hypergeometric …

47. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-47 Hypergeometric Distribution in PHStat Complete dialog box entries and get output … N = 10 n = 3 X = 4 x = 2 P(x = 2) = 0.3 (continued)

48. Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall 5-48 Chapter Summary Reviewed key discrete distributions Binomial Poisson Hypergeometric Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems