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# Math 1300: Section 8-1 Sample Spaces, Events, and Probability

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### Math 1300: Section 8-1 Sample Spaces, Events, and Probability

1. 1. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption Math 1300 Finite MathematicsSection 8-1: Sample Spaces, Events, and Probability Jason Aubrey Department of Mathematics University of Missouri ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
2. 2. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionProbability theory is a branch of mathematics that has beendeveloped do deal with outcomes of random experiments. Arandom experiment (or just experiment) is a situationinvolving chance or probability that leads to results calledoutcomes. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
3. 3. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition The set S of all possible outcomes of an experiment a way that in each trial of the experiment one and only one of the outcomes (events) in the set will occur, we call the set S a sample space for the experiment. Each element in S is called a simple outcome, or simple event. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
4. 4. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition The set S of all possible outcomes of an experiment a way that in each trial of the experiment one and only one of the outcomes (events) in the set will occur, we call the set S a sample space for the experiment. Each element in S is called a simple outcome, or simple event. An event E is deﬁned to be any subset of S (including the empty set and the sample space S). Event E is a simple event if it contains only one element and a compound event if it contains more than one element. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
5. 5. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition The set S of all possible outcomes of an experiment a way that in each trial of the experiment one and only one of the outcomes (events) in the set will occur, we call the set S a sample space for the experiment. Each element in S is called a simple outcome, or simple event. An event E is deﬁned to be any subset of S (including the empty set and the sample space S). Event E is a simple event if it contains only one element and a compound event if it contains more than one element. We say that an event E occurs if any of the simple events in E occurs. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
6. 6. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
7. 7. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) (1, 5) ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
8. 8. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2). (1, 5) (2, 2)What is the sample space S for this experiment? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
9. 9. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(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) ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
10. 10. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionTo clarify, the sample space is always a set of objects. In thiscase,    (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)   ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
11. 11. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionTo clarify, the sample space is always a set of objects. In thiscase,    (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)  We can often use the counting techniques we learned in thelast chapter to determine the size of a sample space. In thiscase, by the multiplication principle: n(S) = 6 × 6 = 36 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
12. 12. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents are subsets of the sample space: ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
13. 13. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents are subsets of the sample space: A simple event is an event (subset) containing only one outcome. For example, E = {(3, 2)} is a simple event. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
14. 14. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents are subsets of the sample space: A simple event is an event (subset) containing only one outcome. For example, E = {(3, 2)} is a simple event. A compound event is an event (subset) containing more than one outcome. For example, E = {(3, 2), (4, 1), (5, 2)} is a compound event. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
15. 15. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents will often be described in words, and the ﬁrst step willbe to determine the correct subset of the sample space. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
16. 16. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents will often be described in words, and the ﬁrst step willbe to determine the correct subset of the sample space. Forexample “A sum of 11 turns up” corresponds to the event E = {(5, 6), (6, 5)}. Notice that n(E) = 2. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
17. 17. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents will often be described in words, and the ﬁrst step willbe to determine the correct subset of the sample space. Forexample “A sum of 11 turns up” corresponds to the event E = {(5, 6), (6, 5)}. Notice that n(E) = 2. “The numbers on the two dice are equal” corresponds to the event F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}. Here we have n(F ) = 6. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
18. 18. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionEvents will often be described in words, and the ﬁrst step willbe to determine the correct subset of the sample space. Forexample “A sum of 11 turns up” corresponds to the event E = {(5, 6), (6, 5)}. Notice that n(E) = 2. “The numbers on the two dice are equal” corresponds to the event F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}. Here we have n(F ) = 6. “A sum less than or equal to 3” corresponds to the event: G = {(1, 1), (1, 2), (2, 1)} ../images/stackedlogo-bw- Here n(G) = 3 Jason Aubrey Math 1300 Finite Mathematics
19. 19. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
20. 20. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.(a) What is the sample space for this experiment? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
21. 21. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.(a) What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
22. 22. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.(a) What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}(b) Identify the event “the outcome is a number greater than15”? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
23. 23. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.(a) What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}(b) Identify the event “the outcome is a number greater than15”? E = {16, 17, 18} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
24. 24. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.(a) What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}(b) Identify the event “the outcome is a number greater than15”? E = {16, 17, 18}(c) Identify the event “the outcome is a number divisible by 12”? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
25. 25. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: A wheel with 18 numbers on the perimeter is spunand allowed to come to rest so that a pointer points within anumbered sector.(a) What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}(b) Identify the event “the outcome is a number greater than15”? E = {16, 17, 18}(c) Identify the event “the outcome is a number divisible by 12”? E = {12} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
26. 26. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionThe ﬁrst step in deﬁning the probability of an event is to assignprobabilities to each of the outcomes (simple events) in thesample space. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
27. 27. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionThe ﬁrst step in deﬁning the probability of an event is to assignprobabilities to each of the outcomes (simple events) in thesample space. Suppose we ﬂip a fair coin twice. The sample space is S = {HH, HT , TH, TT } ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
28. 28. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionThe ﬁrst step in deﬁning the probability of an event is to assignprobabilities to each of the outcomes (simple events) in thesample space. Suppose we ﬂip a fair coin twice. The sample space is S = {HH, HT , TH, TT } Since the coin is fair, each of the four outcomes is equally likely, so P(HH) = P(HT ) = P(TH) = P(TT ) = 1 . 4 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
29. 29. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionSuppose the local meteorologist determines that thechance of rain is 15%. As an experiment, we go out toobserve the weather. The sample space is S = {Rain, No Rain} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
30. 30. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionSuppose the local meteorologist determines that thechance of rain is 15%. As an experiment, we go out toobserve the weather. The sample space is S = {Rain, No Rain}The two outcomes here are not equally likely. We haveP(Rain) = 0.15 and P(No Rain) = 0.85. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
31. 31. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption Suppose the local meteorologist determines that the chance of rain is 15%. As an experiment, we go out to observe the weather. The sample space is S = {Rain, No Rain} The two outcomes here are not equally likely. We have P(Rain) = 0.15 and P(No Rain) = 0.85.Notice that in both cases, each probability was between zeroand one, and the sum of all of the probabilities was one. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
32. 32. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probabilities of Simple Events)Given a sample space S = {e1 , e2 , . . . , en }with n simple events, to each simple event ei we assign a realnumber, denoted by P(ei ), called the probability of the eventei . ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
33. 33. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probabilities of Simple Events)Given a sample space S = {e1 , e2 , . . . , en }with n simple events, to each simple event ei we assign a realnumber, denoted by P(ei ), called the probability of the eventei . 1 The probability of a simple event is a number between 0 and 1, inclusive. That is, 0 ≤ P(ei ) ≤ 1. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
34. 34. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probabilities of Simple Events)Given a sample space S = {e1 , e2 , . . . , en }with n simple events, to each simple event ei we assign a realnumber, denoted by P(ei ), called the probability of the eventei . 1 The probability of a simple event is a number between 0 and 1, inclusive. That is, 0 ≤ P(ei ) ≤ 1. 2 The sum of the probabilities of all simple events in the sample space is 1. That is, P(e1 ) + P(e2 ) + · · · + P(en ) = 1. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
35. 35. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionTwo coin ﬂips. . . A possibly rainy day. . . S = {Rain, No Rain} S = {HH, HT , TH, TT } e P(e) e P(e) Rain 0.15 1 HH 4 No Rain 0.85 1 HT 4 1 TH 4 1 TT 4 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
36. 36. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose a fair coin is ﬂipped twice. What is theprobability that exactly one head turns up. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
37. 37. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose a fair coin is ﬂipped twice. What is theprobability that exactly one head turns up.The event “exactly one head turns up” is the set E = {HT , TH} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
38. 38. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose a fair coin is ﬂipped twice. What is theprobability that exactly one head turns up.The event “exactly one head turns up” is the set E = {HT , TH} 1We know that P(HT ) = 4 and P(TH) = 1 . 4 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
39. 39. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose a fair coin is ﬂipped twice. What is theprobability that exactly one head turns up.The event “exactly one head turns up” is the set E = {HT , TH} 1We know that P(HT ) = 4 and P(TH) = 1 . 4To determine P(E), just add the probabilities of the simpleevents in E. 1 1 1 P(E) = P(HT ) + P(TH) = + = 4 4 2 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
40. 40. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we deﬁne the probability of an arbitraryevent E, denoted by P(E), as follows: ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
41. 41. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we deﬁne the probability of an arbitraryevent E, denoted by P(E), as follows: 1 If E is the empty set, then P(E) = 0. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
42. 42. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we deﬁne the probability of an arbitraryevent E, denoted by P(E), as follows: 1 If E is the empty set, then P(E) = 0. 2 If E is a simple event, i.e. E = {ei }, then P(E) = P(ei ) as deﬁned previously. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
43. 43. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we deﬁne the probability of an arbitraryevent E, denoted by P(E), as follows: 1 If E is the empty set, then P(E) = 0. 2 If E is a simple event, i.e. E = {ei }, then P(E) = P(ei ) as deﬁned previously. 3 If E is a compound event, then P(E) is the sum of the probabilities of all the simple events in E. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
44. 44. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionDeﬁnition (Probability of an Event E)Given a probability assignment for the simple events in asample space S, we deﬁne the probability of an arbitraryevent E, denoted by P(E), as follows: 1 If E is the empty set, then P(E) = 0. 2 If E is a simple event, i.e. E = {ei }, then P(E) = P(ei ) as deﬁned previously. 3 If E is a compound event, then P(E) is the sum of the probabilities of all the simple events in E. 4 If E is the sample space S, then P(E) = P(S) = 1. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
45. 45. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: In a family with 3 children, excluding multiple births,what is the probability of having exactly 2 girls? Assume that aboy is as likely as a girl at each birth. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
46. 46. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: In a family with 3 children, excluding multiple births,what is the probability of having exactly 2 girls? Assume that aboy is as likely as a girl at each birth. First we determine the sample space S: S = {GGG, GGB, GBG, BGG, GBB, BGB, BBG, BBB} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
47. 47. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: In a family with 3 children, excluding multiple births,what is the probability of having exactly 2 girls? Assume that aboy is as likely as a girl at each birth. First we determine the sample space S: S = {GGG, GGB, GBG, BGG, GBB, BGB, BBG, BBB} Since a boy is as likely as a girl at each birth, each of the 8 outcomes in S is equally likely; so each outcome has 1 probability 8 . ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
48. 48. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionNow we identify the event we wish to ﬁnd the probability of: E = {GGB, GBG, BGG} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
49. 49. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionNow we identify the event we wish to ﬁnd the probability of: E = {GGB, GBG, BGG}Therefore, P(E) = P(GGB) + P(GBG) + P(BGG) 1 1 1 3 = + + = 8 8 8 8 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
50. 50. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionProcedure: Steps for Finding the Probability of an Event E ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
51. 51. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionProcedure: Steps for Finding the Probability of an Event E 1 Set up an appropriate sample space S for the experiment. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
52. 52. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionProcedure: Steps for Finding the Probability of an Event E 1 Set up an appropriate sample space S for the experiment. 2 Assign acceptable probabilities to the simple events in S. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
53. 53. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionProcedure: Steps for Finding the Probability of an Event E 1 Set up an appropriate sample space S for the experiment. 2 Assign acceptable probabilities to the simple events in S. 3 To obtain the probability of an arbitrary event E, add the probabilities of the simple events in E. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
54. 54. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption Recall two past examples. . .Two coin ﬂips. . . S = {HH, HT , TH, TT } e P(e) 1 HH 4 1 HT 4 1 TH 4 1 TT 4 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
55. 55. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption Recall two past examples. . .Two coin ﬂips. . . A possibly rainy day. . . S = {Rain, No Rain} S = {HH, HT , TH, TT } e P(e) e P(e) Rain 0.15 1 HH 4 No Rain 0.85 1 HT 4 1 TH 4 1 TT 4 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
56. 56. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption Recall two past examples. . .Two coin ﬂips. . . A possibly rainy day. . . S = {Rain, No Rain} S = {HH, HT , TH, TT } e P(e) e P(e) Rain 0.15 1 HH 4 No Rain 0.85 1 HT 4 1 The outcomes are not equally TH 4 1 likely. TT 4The outcomes are equallylikely. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
57. 57. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionSometimes we can assume that all outcomes in a samplespace are equally likely. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
58. 58. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionSometimes we can assume that all outcomes in a samplespace are equally likely.If S = {e1 , e2 , . . . , en } is a sample space in which alloutcomes are equally likely, then we assign the probability1n to each outcome. That is 1 P(ei ) = nand we have. . . ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
59. 59. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionTheorem (Probability of an Arbitrary Event under an EquallyLikely Assumption)If we assume that each simple event in a sample space S is aslikely to occur as any other, then the probability of an arbitraryevent E in S is given by number of elements in E n(E) P(E) = = . number of elements in S n(S) ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
60. 60. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
61. 61. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) (1, 5) ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
62. 62. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2). (1, 5) (2, 2) ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
63. 63. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: Suppose an experiment consists of simultaneouslyrolling two fair six-sided dice (say, one red die and one greendie) and recording the values on each die. Some possibilitiesare (Red=1, Green=5) or (Red=2, Green=2). (1, 5) (2, 2)(a) What is the probability that the sum on the two dice comesout to be 11? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
64. 64. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionFirstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1, 5), (3, 2), etc, is just as likely as any other. So,we can make an equally likely assumption. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
65. 65. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionFirstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1, 5), (3, 2), etc, is just as likely as any other. So,we can make an equally likely assumption.We know from earlier that n(S) = 36. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
66. 66. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionFirstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1, 5), (3, 2), etc, is just as likely as any other. So,we can make an equally likely assumption.We know from earlier that n(S) = 36.E = {(5, 6), (6, 5)}, so n(E) = 2. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
67. 67. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionFirstly, since the dice are fair, for each die, the numbers 1-6are all equally likely to turn up. So each possible pair ofnumbers (1, 5), (3, 2), etc, is just as likely as any other. So,we can make an equally likely assumption.We know from earlier that n(S) = 36.E = {(5, 6), (6, 5)}, so n(E) = 2.Therefore n(E) 2 1 P(E) = = = n(S) 36 18 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
68. 68. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) What is the probability that the numbers on the dice areequal? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
69. 69. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) What is the probability that the numbers on the dice areequal? The event here is F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
70. 70. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) What is the probability that the numbers on the dice areequal? The event here is F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} So, n(F ) = 6 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
71. 71. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) What is the probability that the numbers on the dice areequal? The event here is F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} So, n(F ) = 6 Therefore, n(F ) 6 1 P(F ) = = = n(S) 36 6 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
72. 72. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: 5 cards are drawn simultaneously from a standarddeck of 52 cards.(a) Describe the sample space S. What is n(S)? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
73. 73. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: 5 cards are drawn simultaneously from a standarddeck of 52 cards.(a) Describe the sample space S. What is n(S)?Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as S = {all possible 5 card hands} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
74. 74. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: 5 cards are drawn simultaneously from a standarddeck of 52 cards.(a) Describe the sample space S. What is n(S)?Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as S = {all possible 5 card hands}How many 5-card hands can be drawn from a 52-card deck? ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
75. 75. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: 5 cards are drawn simultaneously from a standarddeck of 52 cards.(a) Describe the sample space S. What is n(S)?Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as S = {all possible 5 card hands}How many 5-card hands can be drawn from a 52-card deck?From the previous chapter, we know this is n(S) = C(52, 5) = 2, 598, 960 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
76. 76. Experiments Sample Spaces and Events Probability of an Event Equally Likely AssumptionExample: 5 cards are drawn simultaneously from a standarddeck of 52 cards.(a) Describe the sample space S. What is n(S)?Each outcome is a set of 5 cards chosen from the 52 availablecards. So, the sample space S can be described as S = {all possible 5 card hands}How many 5-card hands can be drawn from a 52-card deck?From the previous chapter, we know this is n(S) = C(52, 5) = 2, 598, 960When the cards are dealt, each card is just as likely as anyother, so any ﬁve card hand is just as likely as any other. In ../images/stackedlogo-bw-other words, we can make an equally likely assumption. Jason Aubrey Math 1300 Finite Mathematics
77. 77. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(b)Find the probability that all of the cards are hearts. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
78. 78. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(b)Find the probability that all of the cards are hearts.The event “all of the cards are hearts” is the set E = {all 5 card hands with only hearts} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
79. 79. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(b)Find the probability that all of the cards are hearts.The event “all of the cards are hearts” is the set E = {all 5 card hands with only hearts}Since there are 13 hearts in a standard deck of cards, we have n(E) = C(13, 5) = 1287 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
80. 80. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(b)Find the probability that all of the cards are hearts.The event “all of the cards are hearts” is the set E = {all 5 card hands with only hearts}Since there are 13 hearts in a standard deck of cards, we have n(E) = C(13, 5) = 1287By the equally likely assumption n(E) 1287 P(E) = = ≈ 0.000495 n(S) 2, 598, 960or about 0.05%. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
81. 81. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings). ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
82. 82. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).The event “all the cards are face cards” is the set F = {all 5 card hands consisting only of face cards} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
83. 83. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).The event “all the cards are face cards” is the set F = {all 5 card hands consisting only of face cards}There are a total of 4 × 3 = 12 face cards. So, n(F ) = C(12, 5) = 792 ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
84. 84. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(c) Find the probability that all the cards are face cards (that is,jacks, queens or kings).The event “all the cards are face cards” is the set F = {all 5 card hands consisting only of face cards}There are a total of 4 × 3 = 12 face cards. So, n(F ) = C(12, 5) = 792By the equally likely assumption n(F ) 792 P(F ) = = ≈ 0.000305 n(S) 2, 598, 960 ../images/stackedlogo-bw-or about 0.03%. Jason Aubrey Math 1300 Finite Mathematics
85. 85. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13). ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
86. 86. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).The event “all the cards are even is the set G = {all 5 card hands consisting of only even cards} ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
87. 87. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).The event “all the cards are even is the set G = {all 5 card hands consisting of only even cards}There are 6 even cards per suit (2,4,6,8,10,Q); so there are atotal of 20 even cards in a deck. So, n(G) = C(20, 6) = 38, 760. ../images/stackedlogo-bw- Jason Aubrey Math 1300 Finite Mathematics
88. 88. Experiments Sample Spaces and Events Probability of an Event Equally Likely Assumption(d) Find the probability that all the cards are even. (Consideraces to be 1, jacks to be 11, queens to be 12 and kings to be13).The event “all the cards are even is the set G = {all 5 card hands consisting of only even cards}There are 6 even cards per suit (2,4,6,8,10,Q); so there are atotal of 20 even cards in a deck. So, n(G) = C(20, 6) = 38, 760.By the qually likely assumption, n(G) 38, 760 P(G) = = ≈ 0.0149 n(S) 2, 598, 960 ../images/stackedlogo-bw-or about 14.9%. Jason Aubrey Math 1300 Finite Mathematics