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7 Probability Copyright © Cengage Learning. All rights reserved. 7.1 Sample Spaces and Events Copyright © Cengage Learning. All rights reserved. Sample Spaces 3 Sample Spaces At the beginning of a football game, to ensure fairness, the referee tosses a coin to decide who will get the ball first. When the ref tosses the coin and observes which side faces up, there are two possible results: heads (H) and tails (T). These are the only possible results, ignoring the (remote) possibility that the coin lands on its edge. The act of tossing the coin is an example of an experiment. 4 Sample Spaces The two possible results, H and T, are possible outcomes of the experiment, and the set S = {H, T} of all possible outcomes is the sample space for the experiment. Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment takes place. The set of all possible outcomes is called the sample space for the experiment. 5 Sample Spaces Quick Examples 1. Experiment: Flip a coin and observe the side facing up. Outcomes: H, T Sample Space: S = {H, T} 2. Experiment: Select a student in your class. Outcomes: The students in your class Sample Space: The set of students in your class 3. Experiment: Cast a die and observe the number facing up. Outcomes: 1, 2, 3, 4, 5, 6 Sample Space: S = {1, 2, 3, 4, 5, 6} 6 Sample Spaces 4. Experiment: Cast two distinguishable dice and observe the numbers facing up. Outcomes: (1, 1), (1, 2), . . . , (6, 6) (36 outcomes) Sample S= Space: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) n(S) = the number of outcomes in S = 36 7 Sample Spaces 5. Experiment: Choose 2 cars (without regard to order) at random from a fleet of 10. Outcomes: Collections of 2 cars chosen from 10 Sample Space: The set of all collections of 2 cars chosen from 10 n(S) = C(10, 2) = 45 8 Example 1 – School and Work In a survey conducted by the Bureau of Labor Statistics, the high school graduating class of 2010 was divided into those who went on to college and those who did not. Those who went on to college were further divided into those who went to 2-year colleges and those who went to 4-year colleges. All graduates were also asked whether they were working or not. Find the sample space for the experiment “Select a member of the high school graduating class of 2010 and classify his or her subsequent school and work activity.” 9 Example 1 – Solution The tree in Figure 1 shows the various possibilities. Figure 1 10 Example 1 – Solution cont’d The sample space is S = {2-year college & working, 2-year college & not working, 4-year college & working, 4-year college & not working, no college & working, no college & not working}. 11 Events 12 Events In Example 1, suppose we are interested in the event that a 2010 high school graduate was working. In mathematical language, we are interested in the subset of the sample space consisting of all outcomes in which the graduate was working. Events Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E—that is, if the outcome of the experiment is favorable. 13 Events Visualizing an Event In the following figure, the favorable outcomes (events in E) are shown in green. 14 Events Quick Examples 1. Experiment: Roll a die and observe the number facing up. S = {1, 2, 3, 4, 5, 6} Event: E: The number observed is odd. E = {1, 3, 5} 2. Experiment: Roll two distinguishable dice and observe the numbers facing up. S = {(1, 1), (1, 2), . . . , (6, 6)} Event: F: The dice show the same number. F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} 15 Events 3. Experiment: Roll two distinguishable dice and observe the numbers facing up. S = {(1, 1), (1, 2), . . . , (6, 6)} Event: G: The sum of the numbers is 1. There are no favorable outcomes. G=∅ 4. Experiment: Select a city beginning with “J.” Event: E: The city is Johannesburg. E = {Johannesburg} An event can consist of a single outcome. 16 Events 5. Experiment: Draw a hand of 2 cards from a deck of 52. Event: H: Both cards are diamonds. H is the set of all hands of 2 cards chosen from 52 such that both cards are diamonds. 17 Example 2 – Dice We roll a red die and a green die and observe the numbers facing up. Describe the following events as subsets of the sample space. a. E: The sum of the numbers showing is 6. b. F: The sum of the numbers showing is 2. 18 Example 2 – Solution Here (again) is the sample space for the experiment of throwing two dice. (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), S= (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) 19 Example 2 – Solution cont’d a. In mathematical language, E is the subset of S that consists of all those outcomes in which the sum of the numbers showing is 6. Here is the sample space once again, with the outcomes in question shown in color: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), S= (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) Thus, E = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}. 20 Example 2 – Solution cont’d b. The only outcome in which the numbers showing add to 2 is (1, 1). Thus, F = {(1, 1)}. 21 Complement, Union, and Intersection of Events 22 Complement, Union, and Intersection of Events Events may often be described in terms of other events, using set operations such as complement, union, and intersection. Complement of an Event The complement of an event E is the set of outcomes not in E. Thus, the complement of E represents the event that E does not occur. Visualizing the Complement 23 Complement, Union, and Intersection of Events Quick Examples 1. You take four shots at the goal during a soccer game and record the number of times you score. Describe the event that you score at least twice, and also its complement. S = {0, 1, 2, 3, 4} Set of outcomes E = {2, 3, 4} Event that you score at least twice E = {0, 1} Event that you do not score at least twice 24 Complement, Union, and Intersection of Events 2. You roll a red die and a green die and observe the two numbers facing up. Describe the event that the sum of the numbers is not 6. S = {(1, 1), (1, 2), . . . , (6, 6)} F = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)} F = (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1), (4, 3), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3), Sum of numbers is 6. (1, 4), (3, 4), (4, 4), (5, 4), (6, 4), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6) Sum of numbers is not 6. 25 Complement, Union, and Intersection of Events Union of Events The union of the events E and F is the set of all outcomes in E or F (or both). Thus, E U F represents the event that E occurs or F occurs (or both). Quick Example Roll a die. E: The outcome is a 5; E = {5}. F: The outcome is an even number; F = {2, 4, 6}. E U F: The outcome is either a 5 or an even number; E U F = {2, 4, 5, 6}. 26 Complement, Union, and Intersection of Events Intersection of Events The intersection of the events E and F is the set of all outcomes common to E and F. Thus, E ∩ F represents the event that both E and F occur. Quick Example Roll two dice; one red and one green. E: The red die is 2. F: The green die is odd. E ∩ F: The red die is 2 and the green die is odd; E ∩ F = {(2, 1), (2, 3), (2, 5)}. 27 Example 4 – Weather Let R be the event that it will rain tomorrow, let P be the event that it will be pleasant, let C be the event that it will be cold, and let H be the event that it will be hot. a. Express in words: R ∩ P, R U (P ∩ C). b. Express in symbols: Tomorrow will be either a pleasant day or a cold and rainy day; it will not, however, be hot. Solution: The key here is to remember that intersection corresponds to and and union to or. 28 Example 4 – Solution cont’d a. R ∩ P is the event that it will rain and it will not be pleasant. R U (P ∩ C) is the event that either it will rain, or it will be pleasant and cold. b. If we rephrase the given statement using and and or we get “Tomorrow will be either a pleasant day or a cold and rainy day, and it will not be hot.” [P U (C ∩ R)] ∩ H Pleasant, or cold and rainy, and not hot. 29 Example 4 – Solution cont’d The nuances of the English language play an important role in this formulation. For instance, the effect of the pause (comma) after “rainy day” suggests placing the preceding clause P U (C ∩ R) in parentheses. In addition, the phrase “cold and rainy” suggests that C and R should be grouped together in their own parentheses. 30 Complement, Union, and Intersection of Events The case where E ∩ F is empty is interesting, and we give it a name. Mutually Exclusive Events If E and F are events, then E and F are said to be disjoint or mutually exclusive if E ∩ F is empty. (Hence, they have no outcomes in common.) Visualizing Mutually Exclusive Events 31 Complement, Union, and Intersection of Events Interpretation It is impossible for mutually exclusive events to occur simultaneously. Quick Examples In each of the following examples, E and F are mutually exclusive events. 1. Roll a die and observe the number facing up. E: The outcome is even; F: The outcome is odd. E = {2, 4, 6}, F = {1, 3, 5} 32 Complement, Union, and Intersection of Events 2. Toss a coin three times and record the sequence of heads and tails. E: All three tosses land the same way up, F: One toss shows heads and the other two show tails. E = {HHH, TTT}, F = {HTT, THT, TTH} 3. Observe tomorrow’s weather. E: It is raining; F: There is not a cloud in the sky. 33