1. Chapter 5 Probability

2. Chance experiment – any activity or situation in which there is uncertainty about which of two or more plausible outcomes will result. Suppose two six-sided dice are rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection. These would be examples of chance experiments.

3. Sample space - the collection of all possible outcomes of a chance experiment Suppose a six-sided die is rolled. The possible outcomes are that the die could land with 1 dot up or 2, 3, 4, 5, or 6 dots up. S = {1, 2, 3, 4, 5, 6} This would be an example of a sample space. “S” stands for sample space. We use set notation to list the outcomes of the sample space. The sum of the probabilities of the outcomes in the sample space equals ONE.

4. Suppose two coins are flipped. The sample space would be: S = {HH, HT, TH, TT} Where H = heads and T = tails H T H T H T We can also use a tree diagram to represent a sample space. HT We follow the branches out to show an outcome.

5. Event - any collection of outcomes (subset) from the sample space of a chance experiment Suppose a six-sided die is rolled. The outcome that the die would land on an even number would be E = {2, 4, 6} This would be an example of an event. We typically use capital letters to denote an event.

6. Complement - Consists of all outcomes that are not in the event Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} What would the event be that is the die NOT landing on an even number? E C = {1, 3, 5} This is an example of complementary events. The superscript “C” stands for complement E’ and ~E also denote the complement of E The sum of the probabilities of complementary events equals ONE.

7. Probability with Two-Way Tables

8. The article “Chances Are You Know Someone with a Tattoo, and He’s Not a Sailor” (Associated Press, June 11, 2006) included results from a survey of adults aged 18 to 50. The accompanying data are consistent with summary values given in the article. At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 Assuming these data are representative of adult Americans and that an adult is selected at random, use the given information to estimate the following probabilities.

9. Tattoo Example Continued... At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 What is the probability that a randomly selected adult has a tattoo? P(tattoo) =24/100 = 0.24

10. Tattoo Example Continued... At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 What is the probability that a randomly selected adult has a tattoo and is age 18 to 29? P(tattoo) =18/100 = 0.18

11. Tattoo Example Continued... At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 What is the probability that a randomly selected adult has a tattoo or is age 18 to 29? P(tattoo or age 18-29) = (24 + 50 – 18)/100 = 0.56 Note that “age 16-29 and at least one tattoo” or 18 is counted twice!

12. Tattoo Example Continued... At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 What is the probability that a randomly selected adult has a tattoo if they are between 18 and 29 years old? P(tattoo|age 18-29) =18/50 = 0.36 This is a condition! How many adults in the sample are ages 18- 29? How many adults in the sample are ages 18-29 AND have a tattoo?

13. Tattoo Example Continued... At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 What is the probability that a randomly selected adult is between 18 and 29 years old if they have a tattoo? P(age 18-29|tattoo) =18/24 = 0.75 This is a condition! How many adults in the sample have a tattoo? How many adults in the sample are ages 18-29 AND have a tattoo?

14. Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items. Construct a tree diagram or a chart for this data. What is the probability that an item is returned? If an item is returned, what is the probability that an inexperienced employee assembled it? P(returned) = 4.8/100 = 0.048 P(inexperienced|returned) = 2.4/4.8 = 0.5

15. These complementary events can be shown on a Venn Diagram. E = {2, 4, 6} and E C = {1, 3, 5} Let the rectangle represent the sample space. Let the circle represent event E. Let the shaded area represent event not E. 2 4 6 1 3 5

16. Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E or P happening? E or P = {2, 3, 4, 5, 6} This is an example of the union of two events.

17. The union of A or B - consists of all outcomes that are in at least one of the two events, that is, in A or in B or in both. This symbol means “union” Consider a marriage or union of two people – when two people marry, what do they do with their possessions ? The bride takes all her stuff & the groom takes all his stuff & they put it together! And live happily ever after! This is similar to the union of A and B. All of A and all of B are put together!

18. Let’s revisit rolling a die and getting an even or a prime number... E or P = {2, 3, 4, 5, 6} Another way to represent this is with a Venn Diagram. Even number 2 4 6 Prime number 3 5 1 E or P would be any number in either circle. Why is the number 1 outside the circles?

19. Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E and P happening? E and P = {2} This is an example of the intersection of two events.

20. The intersection of A and B - consists of all outcomes that are in both of the events This symbol means “intersection”

21. Let’s revisit rolling a die and getting an even or a prime number... E and P = {2} To represent this with a Venn Diagram: 2 4 6 3 5 1 E and P would be ONLY the middle part that the circles have in common

22. Mutually exclusive (or disjoint) events - two events have no outcomes in common; two events that NEVER happen simultaneously Suppose a six-sided die is rolled. Consider the following 2 events: A = {2} B = {6} On a single die roll, is it possible for A and B to happen at the same time? These events are mutually exclusive.

23. A Venn Diagram for the roll of a six-sided die and the following two events: A = {2} B = {6} 2 4 6 3 5 1 A and B are mutually exclusive (disjoint) since they have no outcomes in common The intersection of A and B is empty!

24. Independent Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs Think about your high school... Suppose a student is selected at random. What is the probability that the student plays basketball? Does knowing that that student is female change that probability?

25. Independent Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs Think about your high school... Suppose a student is selected at random. What is the probability that the student plays football? Does knowing that that student is female change that probability? If two events are not independent, they are said to be dependent events.

26. What is Probability? Two different approaches to probability

27. The Classical Approach When the outcomes in a sample space are equally likely, the probability of an event E, denoted by P(E), is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space. Examples: flipping a coin, rolling a die, etc.

28. On some football teams, the honor of calling the toss at the beginning of the football game is determined by random selection. Suppose this week a member of the 11-player offensive team will be selected to call the toss. There are five interior linemen on the offensive team. If event L is defined as the event that an interior linemen is selected to call the toss, what is probability of L? P(L) = 5/11

29. Consider an archer shooting arrows at a target. The probability of getting a bulls’ eye should be the ratio of the area of the inner circle to the area of the entire target. What if a very experienced archer were shooting the arrows? Would the probability of a bull’s eye still be the same? The classical approach doesn’t work for every situation.

30. The Relative Frequency Approach The probability of event E, denoted by P(E), is defined to be the value approached by the relative frequency of occurrence of E in a very long series of trials of a chance experiment. Thus, if the number of trials is quite large,

31. Consider flipping a coin and recording the relative frequency of heads. When the number of coin flips is small, there is a lot of variability in the relative frequency of “heads” (as shown in this graph). What do you notice in the graph at the right?

32. Consider flipping a coin and recording the relative frequency of heads. The graph at the right shows the relative frequency when the coin is flipped a large number of times. What do you notice in this graph at the right?

33. Law of Large Numbers As the number of repetitions of a chance experiment increase, the chance that the relative frequency of occurrence for an event will differ from the true probability by more than any small number approaches 0. OR in other words, after a large number of trials, the relative frequency approaches the true probability. Notice how the relative frequency of heads approaches ½ the larger the number of trials!

34. Law of Averages?? There is NOT a Law of Averages! Students often try to extend the Law of Large Numbers to averages. The Law of Large Numbers is about PROBABILITY!!

35. Probability Rules!

36. Fundamental Rules of Probability Rule 1. Legitimate Values For any event E, 0 < P(E) < 1 Rule 2. Sample space If S is the sample space, P(S) = 1

37. Rules Continued... Rule 3. Complement For any event E, P(E) + P(E C ) = 1

38. Rule 4: Multiplication Two events E and F are independent, if and only if, Rules Continued...

39. Let E 1 = event that a newly purchased monitor is not defective E 2 = event that a newly purchased mouse is not defective E 3 = event that a newly purchased disk drive is not defective E 4 = event that a newly purchased processor is not defective Suppose the four events are independent with P(E 1 ) = P(E 2 ) =.98 P(E 3 ) =.94 P(E 4 ) =.99 What is the probability that none of these components are defective? (.98)(.98)(.94)(.99) =.89 In the long run, 89% of such systems will run properly when tested shortly after purchase.

40. Suppose the manufacturer of a certain brand of light bulbs made 10,000 of these bulbs and 500 are defective. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that both bulbs are defective? Are the events E 1 = the first bulb is defective and E 2 = the second bulb is defective independent? What would be the probability of selecting a defective light bulb? 500/10,000 =.05 To answer this question, let’s explore the probabilities of these two events?

41. Light Bulbs Continued... What would be the probability of selecting a defective light bulb? Having selected one defective bulb, what is the probability of selecting another without replacement? 500/10,000 =.05 499/9999 =.0499 These values are so close to each other that when rounded to three decimal places they are both.050. For all practical purposes, we can treat them as being independent. If a random sample of size n is taken from a population of size N, then the outcomes of selecting successive items from the population without replacement can be treated as independent when the sample size n is at most 10% of the population size N.

42. Light Bulbs Continued... What is the probability that both bulbs are defective? Are the selections independent? We can assume independence. (0.05)(0.05) =.0025

43. Light Bulbs Revisited... A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that exactly one bulb is defective? Let D 1 = first light bulb is defective D 2 = second light bulb is defective = (.05)(.95) + (.95)(.05) =.095

44. Rules Continued... Rule 4 Revisited: General Multiplication For any two events E and F,

45. There are seven girls and eight boys in a math class. The teacher selects two students at random to answer questions on the board. What is the probability that both students are girls? Are these events independent? NO

46. Rule 5: Additive Two events E and F are disjoint, if and only if, Rules Continued...

47. A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. Consider a chance experiment that consist of observing the make of the next car sold. Suppose that P(H) = 0.25, P(N) = 0.18, P(T) = 0.14. Are these disjoint events? P(H or N or T) = P(not (H or N or T)) = yes.25 +.18+.14 =.57 1 -.57 =.43

48. Rules Continued... Rule 5 Revisited: General Addition For any two events E and F, EF Since the intersection is added in twice, we subtract out the intersection.

49. Musical styles other than rock and pop are becoming more popular. A survey of college students finds that the probability they like country music is.40. The probability that they liked jazz is.30 and that they liked both is.10. What is the probability that they like country or jazz?.4 +.3 -.1 =.6

50. Ask yourself, “Are the events mutually exclusive?” Yes No If independent Here is a process to use when calculating the union of two or more events. In some problems, the intersection of the two events is given (see previous example). In some problems, the intersection of the two events is not given, but we know that the events are independent.

51. Suppose two six-sided dice are rolled (one white and one red). What is the probability that the white die lands on 6 or the red die lands on 1? Let A = white die landing on 6 B = red die landing on 1 Are A and B disjoint? NO, independent events cannot be disjoint How can you find the probability of A and B?

52. Rules Continued... Rule 6: Conditional Probability A probability that takes into account a given condition has occurred Think:

53. Suppose I will pick two cards from a standard deck. This can be done two ways: 1)Pick a card at random, replace the card, then pick a second card 2) Pick a card at random, do NOT replace, then pick a second card. If I pick two cards from a standard deck without replacement, what is the probability that I select two spades? Are the events S 1 = first card is a spade and S 2 = second card is a spade independent? NO P(S 1 and S 2 ) = P(S 1 ) × P(S 2 |S 1 ) = Sampling with replacement – the events are typically independent events. Sampling without replacement – the events are typically dependent events. Probability of a spade given I drew a spade on the first card.

54. In a recent study it was found that the probability that a randomly selected student is a girl is.51 and is a girl and plays sports is.10. If the student is female, what is the probability that she plays sports? This is the condition – it goes on the bottom! The “and” of these goes on top

55. Ask yourself, “ Are these events independent?” Yes No Here is a process to use when calculating the intersection of two or more events.

56. Tests for independence There are two ways to test for independence 1.Use the multiplication rule 2.Use the conditional rule

57. Tattoo Example Continued... At Least One Tattoo No TattooTotals Age 18-29183250 Age 30-5064450 Totals2476100 Is the event “Having a tattoo” independent of being “Age 18-29” P(tattoo) =24/100 = 0.24 P(tattoo| age 18 - 29) = 18/50 = 0.36 Since these probabilities are not equal, these events are NOT independent!

58. Extra Problems The American Red Cross must track their supply and demand of various blood types. They estimate that about 45% of the U.S. population has Type O, 40% Type A, 11% Type B, and the rest Type AB. If someone volunteers to give blood, what is the probability that this donor: a)Has Type AB blood? b)Has Type A or Type B blood? c)Is not Type O? 0.04 0.85 0.55

59. Extra Problems The American Red Cross must track their supply and demand of various blood types. They estimate that about 45% of the U.S. population has Type O, 40% Type A, 11% Type B, and the rest Type AB. Among four potential donors, what is the probability that: a)All are Type O? b)None have Type AB? c)Not all are Type A? d)At least one person is Type B? 0.041 0.849 0.9744 0.373

60. Extra Problems Because gambling is big business, calculating the odds of a gambler winning or losing in every game is crucial to the financial forecasting for a casino. A standard lot machine has three wheels that spin independently. Each has 10 equally likely symbols: 4 bars, 3 lemons, 2 cherries, and a bell. If you play once, what is the probability that you will get: a)3 lemons b)No fruit symbol c)3 bells (jackpot) d)No bells e)At least one bar (automatic loser) 0.125 0.001 0.729 0.784 0.027