1. College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson

2. Counting and Probability 10

3. Probability 10.3

4. Overview In the preceding chapters, we modeled real-world situations. These were modeled using precise rules, such equations or functions.

5. Overview However, many of our everyday activities are not governed by precise rules. Rather, they involve randomness and uncertainty.

6. Overview How can we model such situations? Also, how can we find reliable patterns in random events? In this section, we will see how the ideas of probability provide answers to these questions.

7. Rolling a Die Let’s look at a simple example. We roll a die, and we’re hoping to get a “two”. Of course, it’s impossible to predict what number will show up.

8. Rolling a Die But, here’s the key idea: We roll the die many many times. Then, the number two will show up about one-sixth of the time.

9. Rolling a Die This is because each of the six numbers is equally likely to show up. So, the “two” will show up about a sixth of the time. If you try this experiment, you will see that it actually works!

10. Rolling a Die We say that the probability (or chance) of getting “two” is 1/6.

11. Picking a Card If we pick a card from a 52-card deck, what are the chances that it is an ace? Again, each card is equally likely to be picked. Since there are four aces, the probability (or chances) of picking an ace is 4/52.

12. Probability and Science Probability plays a key role in many sciences. A remarkable example of the use of probability is Gregor Mendel’s discovery of genes. He could not see the genes. His discovery was due to applying probabilistic reasoning to the patterns he saw in inherited traits.

13. Probability Today, probability is an indispensable tool for decision making. It is used in business, industry, government, and scientific research. For example, probability is used to –Determine the effectiveness of new medicine –Assess fair prices for insurance policies –Gauge public opinion on a topic (without interviewing everyone)

14. Probability In the remaining sections of this chapter, we will see how some of these applications are possible.

15. What is Probability?

16. Terminology To discuss probability, let’s begin by defining some terms. An experiment is a process, such as tossing a coin or rolling a die. The experiment gives definite results called the outcomes of the experiment. –For tossing a coin, the possible outcomes are “heads” and “tails” –For rolling a die, the outcomes are 1, 2, 3, 4, 5, and 6.

17. Terminology The sample space of an experiment is the set of all possible outcomes. If we let H stand for heads and T for tails, then the sample space of the coin-tossing experiment is S = {H, T}.

18. Sample Space The table lists some experiments and the corresponding sample spaces.

19. Experiments with Equally Likely Outcomes We will be concerned only with experiments for which all the outcomes are equally likely. We already have an intuitive feeling for what this means. When we toss a perfectly balanced coin, heads and tails are equally likely outcomes. This is in the sense, that if this experiment is repeated many times, we expect that about as many heads as tails will show up.

20. Experiments and Outcomes In any given experiment, we are often concerned with a particular set of outcomes. We might be interested in a die showing an even number. Or, we might be interested in picking an ace from a deck of cards. Any particular set of outcomes is a subset of the sample space.

21. An Event—Definition This leads to the following definition. If S is the sample space of an experiment, then an event is any subset of the sample space.

22. E.g. 1—Events in a Sample Space An experiment consists of tossing a coin three times and recording the results in order. The sample space is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

23. E.g. 1—Events in a Sample Space The event E of showing “exactly two heads” is the subset of S. E consists of all outcomes with two heads. Thus, E = {HHT, HTH, THH}

24. E.g. 1—Events in a Sample Space The event F of showing “at least two heads” is F = {HHH, HHT, HTH, THH} And, the event of showing “no heads” is G = {TTT}

25. Intuitive Notion of Probability We are now ready to define the notion of probability. Intuitively, we know that rolling a die may result in any of six equally likely outcomes. So, the chance of any particular outcome occurring is 1/6.

26. Intuitive Notion of Probability What is the chance of showing an even number? Of the six equally likely outcomes possible, three are even numbers. So it is reasonable to say that the chance of showing an even number is 3/6 = 1/2. This reasoning is the intuitive basis for the following definition of probability.

27. Probability—Definition Let S be the sample space of an experiment in which all outcomes are equally likely. Let E be an event. The probability of E, written P(E), is

28. Values of a Probability Notice that 0 ≤ n(E) ≤ n(S). So, the probability P(E) of an event is a number between 0 and 1. That is, 0 ≤ P(E) ≤ 1. The closer the probability is to 1, the more likely the event is to happen. The close to 0, the less likely.

29. Values of a Probability If P(E) = 1, then E is called the certain event. If P(E) = 0, then E is called the impossible event.

30. E.g. 2—Finding the Probability of an Event A coin is tossed three times, and the results are recorded. What is the probability of getting exactly two heads? At least two heads? No heads?

31. E.g. 2—Finding the Probability of an Event By the results of Example 1, the sample space S of this experiment contains eight outcomes. The event E of getting “exactly two heads” contains three outcomes. They are {HHT, HTH, THH}. So, by the definition of probability,

32. E.g. 2—Finding the Probability of an Event Similarly, the event F of getting “at least two heads” has four outcomes. They are {HHH, HHT, HTH, THH}. So,

33. E.g. 2—Finding the Probability of an Event The event G of getting “no heads” has one element, so

34. Calculating Probability by Counting

35. To find the probability of an event: We do not need to list all the elements in the sample space and the event. What we do need is the number of elements in these sets. The counting techniques that we learned in the preceding sections will be very useful here.

36. E.g. 3—Finding the Probability of an Event A five-card poker hand is drawn from a standard 52-card deck. What is the probability that all five cards are spades? The experiment here consists of choosing five cards from the deck. The sample space S consists of all possible five-card hands.

37. E.g. 3—Finding the Probability of an Event Thus, the number of elements in the sample space is

38. E.g. 3—Finding the Probability of an Event The event E that we are interested in consists of choosing five spades. Since the deck contains only 13 spades, the number of ways of choosing five spades is

39. E.g. 3—Finding the Probability of an Event Thus, the probability of drawing five spades is

40. Understanding a Probability What does the answer to Example 3 tell us? Since 0.0005 = 1/2000, this means that if you play poker many, many times, on average you will be dealt a hand consisting of only spades about once every 2000 hands.

41. E.g. 4—Finding the Probability of an Event A bag contains 20 tennis balls. Four of the balls are defective. If two balls are selected at random from the bag, what is the probability that both are defective?

42. E.g. 4—Finding the Probability of an Event The experiment consists of choosing two balls from 20. So, the number of elements in the sample space S is C(20, 2). Since there are four defective balls, the number of ways of picking two defective balls is C(4, 2).

43. E.g. 4—Finding the Probability of an Event Thus, the probability of the event E of picking two defective balls is

44. Complement of an Event The complement of an event E is the set of outcomes in the sample space that is not in E. We denote the complement of an event E by E′.

45. Complement of an Event We can calculate the probability of E′ using the definition and the fact that n(E′) = n(S) – n(E) So, we have

46. Probability of the Complement of an Event Let S be the sample space of an experiment, and E and event. Then

47. Probability of the Complement of an Event This is an extremely useful result. It is often difficult to calculate the probability of an event E. But, it is easy to find the probability of E′. From this, P(E) can be calculated immediately by using this formula.

48. E.g. 5—The Probability of the Complement of an Event An urn contains 10 red balls and 15 blue balls. Six balls are drawn at random from the urn. What is the probability that at least one ball is red?

49. E.g. 5—The Probability of the Complement of an Event Let E be the event that at least one red ball is drawn. It is tedious to count all the possible ways in which one or more of the balls drawn are red. So let’s consider E′, the complement of this event. E′ is the event that none of the balls drawn are red.

50. E.g. 5—The Probability of the Complement of an Event The number of ways of choosing 6 blue balls from the 15 balls is C(15, 6). The number of ways of choosing 6 balls from the 25 ball is C(25, 6). Thus,

51. E.g. 5—The Probability of the Complement of an Event By the formula for the complement of an event, we have

52. The Union of Events

53. If E and F are events, what is the probability that E or F occurs? The word or indicates that we want the probability of the union of these events. That is,.

54. The Union of Events So, we need to find the number of elements in. If we simply add the number of element in E to the number of elements in F, then we would be counting the elements in the overlap twice. Once in E and once in F.

55. The Union of Events So to get the correct total, we must subtract the number of elements in. Thus,

56. The Union of Events Using the formula for probability, we get We have just proved the following.

57. The Probability of The Union of Events If E and F are events in a sample space S, then the probability of E or F is

58. E.g. 6—The Probability of the Union of an Event What is the probability that a card drawn at random from a standard 52-card deck is either a face card or a spade? We let E and F denote the following events: E: The card is a face card. F: The card is a spade.

59. E.g. 6—The Probability of the Union of an Event There are 12 face cards and 13 spades in a 51-card deck, so

60. E.g. 6—The Probability of the Union of an Event Since three cards are simultaneously face cards and spades, we have

61. E.g. 6—The Probability of the Union of an Event Thus, by the formula for the probability of the union of two events, we have

62. The Union of Mutually Exclusive Events

63. Two events that have no outcome in common are said to be mutually exclusive. This is illustrated in the figure.

64. A Mutually Exclusive Event For example, draw a card from a deck. Consider the events E:The card is an ace. F:The card is a queen. They are mutually exclusive because a card cannot be both an ace and a queen.

65. Mutually Exclusive Events If E and F are mutually exclusive events, then contains no elements. Thus, So, We have proved the following formula.

66. Probability of the Union of Mutually Exclusive Events If E and F are mutually exclusive events in a sample space S, then the probability of E or F is

67. Multiple Mutually Exclusive Events There is a natural extension of this formula for any number of mutually exclusive events: If E 1, E 2, …, E n are pairwise mutually exclusive, then

68. E.g. 7—The Union of Mutually Exclusive Events A card is drawn at random from a standard deck of 52 cards. What is the probability that the card is either a seven or a face card? Let E and F denote the following events: E:The card is a seven. F:The card is a face card.

69. E.g. 7—The Union of Mutually Exclusive Events A card cannot be both a seven and a face card. Thus, the events are mutually exclusive.

70. E.g. 7—The Union of Mutually Exclusive Events We want the probability of E or F. In other words, the probability of.

71. E.g. 7—The Union of Mutually Exclusive Events By the formula,

72. The Intersections of Independent Events

73. The Intersection of Events We have considered the probability of events joined by the word or. That is, the union of events. Now, we study the probability of events joined by the word and. –In other words, the intersection of events.

74. The Intersection of Independent Events When the occurrence of one event does not affect the probability of another event: We say that the events are independent. For instance, if a balanced coin is tossed, the probability of showing heads on the second toss is 1/2. –This is regardless of the outcome of the first toss. –So, any two tosses of a coin are independent.

75. Probability of the Intersection of Independent Events If E and F are independent events in a sample space S, then the probability of E and F is

76. E.g. 8—The Probability of Independent Events A jar contains five red balls and four black balls. A ball is drawn at random from the jar and then replaced. Then, another ball is picked. What is the probability that both balls are red?

77. E.g. 8—The Probability of Independent Events The events are independent. The probability that the first ball is red is 5/9. The probability that the second ball is red is also 5/9. Thus, the probability that both balls are red is

78. E.g. 9—The Birthday Problem What is the probability that in a class of 35 students, at least two have the same birthdays? It is reasonable to assume that the 35 birthdays are independent. It can also be assumed that each day of the 365 days in a year is equally likely as a date of birth. –We ignore February 29.

79. E.g. 9—The Birthday Problem Let E be the event that two of the students have the same birthday. It is tedious to list all the possible ways in which at least two of the students have matching birthdays. So, we consider the complementary event E′. That is, that no two students have the same birthday.

80. E.g. 9—The Birthday Problem To find this probability, we consider the students one at a time. The probability that the first student has a birthday is 1. The probability that the second has a birthday different from the first is 364/365. The probability that the third has a birthday different from the first two is 363/365. And so on.

81. E.g. 9—The Birthday Problem Thus, So,

82. The Birthday Paradox Most people are surprised that the probability in Example 9 is so high. For this reason, this problem is sometimes called the “birthday paradox”.

83. The Birthday Paradox The table below gives the probability that two people in a group will share the same birthday for groups of various sizes.