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Raja Dr. Latif. Math 131 (043) Sec8.3 Pg:1 8.3: Sample Spaces and Events EXPERIMENT An experiment is an activity with observable results. SAMPLE POINT : an outcome of an experiment. SAMPLE SPACE : the set consisting of all possible sample points of an experiment. EVENT : a subset of a sample space of an experiment. UNION OF TWO EVENTS The union of the two events E and F is the event E F. 1 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:2 E F s S : s E or s F . INTERSECTION OF TWO EVENTS The inter sec tion of the two events E and F is the event E F. EF s S : s E&s F . COMPLEMENT OF AN EVENT The complement of an event E is the event Ec. E c s S : s E S E. MUTUALLY EXCLUSIVE EVENTS E and F are mutually exclusive if E F . DeMORGAN'S LAWS: E F c E c F c 2 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:3 E F c E c F c . (True for any number of events) In Exercise 1 6, let S a, b, c, d, e, f be a sample space of an experiment and let E a, b, F a, d, f, and G b, c, e be events of this experiment. 1.Find the events E F and E F. E F a, b, d, f; E F a. 2.Find the events F G and F G. F G a, b, c, d, e, f; FG 3.Find the events F c and F G c . 3 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:4 F c b, c, e; E G c a, b a, d, f a 4.Find the events E c and F c G. E c c, d, e, f; F c G b, c, e b, c, e b, c, e. 5.Are the events E and F mutually exclusive? Since E F a is not a null set, we conclude that E and F are not mutually exclusive. 6.Are the events E F and E F c mutually exclusive? E F a, b, d, f and E F c a, b b, c, e b. 4 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:5 Since b is an element of both sets, they are not mutually exclusive. In Exercise 7 12, let S 1, 2, 3, 4, 5, 6, E 2, 4, 6 F 1, 3, 5, and G 5, 6. 7.Find the event E F G. E F G 2, 4, 6 1, 3, 5 5, 6 1, 2, 3, 4, 5, 6 8.Find the event E F G. EFG 2, 4, 6 1, 3, 5 5, 6 . 9.Find the event E F G c . E F G c 1, 2, 3, 4, 5, 6 c . 5 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:6 10.Find the event E F G c . E F G c 1, 2, 3, 4, 5, 6. 11.Are the events E and F mutually exclusive? Yes, E F ; that is, E and F, do not contain any common elements. 12.Are the events F and G mutually exclusive? No. 5 is an element of both sets. In Exercises 13 18, let S be any sample space in E, F, and G be any three events associated with the experiment. Describe the given events using the 6 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:7 symbols , , and c . 13. The event that E and/or F occurs EF 14. The event that both E and F occur EF 15. The event that G does not occur Gc 16. The event that E but not F occurs E F c 17. The event that none of the events E, F, and G occurs E F G c 18. The events that E occurs but neither of the events F or G occurs. 7 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:8 E F c G c 19. Let S a, b, c be a sample space of an experiment with outcomes a, b, and c. List all the events of this experiments. , a, b, c, a, b, a, c, b, c, a, b, c. 20. Let S 1, 2, 3 be a sample space associated with an experiment. a. List all events of this experiment. a. , 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3 b. How many subsets of S contain the number 3? b. 4 c. How many subsets of S contain either the number 2 or the number 3? 8 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:9 c. 4 2.391TAN22. An experiment consists of selecting a letter at random from the letters in the word MASSACHUSETTS and observing the outcomes. a. What is an appropriate sample space for this experiment. a. A, C, E, H, M, S, T, U b. Describe the event "a head is tossed and an even number is cost." b. A, E, U 3.391TAN23. An experiment consists of tossing a coin and casting a die and observing the outcomes. a. Describe an appropriate sample space 9 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:10 for this experiment. a. S H, 1, H, 2, H, 3, H, 4, H, 5, H, 6, T, 1, T, 2, T, 3, T, 4, T, 5, T, 6. b. Describe the event head "a is tossed and an even number is cast." b. E H, 2, H, 4, H, 6. 4. 391TAN25. Quality Control A sample of three transistors taken from a local electronics store was examined to determine whether the transistors were effective d or non-defective n. What is an appropriate sample space for this experiment? S d, d, d, d, d, n, d, n, d, n, d, d, d, n, n, n, d, n, n, n, d, n, n, n 10 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:11 5.391TAN27. Game Shows In a television game show, the winner is asked to select three prizes from five different prizes, A, B, C, D, and E. a. Describe a sample space for possible outcomes(order is not important). a.ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. b. How many points are there in the sample space corresponding to a selection that includes A. b. 6 c. How many points are there in the sample space corresponding to a selection that include A and B. c. 3 11 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:12 d. How many points are there in the sample space corresponding to a selection that includes either A or B. d. 6 6.392TAN35. Shutle Bus Usage A certain airport hotel operates a shuttle bus service between the hotel and the airport. The maximum capacity of a bus is 20 passengers. On alternate trips of the shuttle bus over a period of 1 week, the hotel manager kept a record of the number of passengers arriving at the hotel in each bus. a. What is an appropriate sample space for this experiment? a. S 0, 1, 2, . . . . . , 20 b. Describe the event E that a shuttle bus carried fewer than ten passengers. 12 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:13 b. E 0, 1, 2, . . . , 9 c. Describe the event F that a shuttle bus arrived with a full capacity. c. F 20 7.393TAN38. Let S be a sample space for an experiment and let E and F be events of this experiment. Show that the events E F and E c F c are mutually exclusive. Hint : Use De Morgan , s law. E c F c E F c By De Morgans law. Since E F E F c , they are mutually exclusive. 8.393TAN39. Let S be a sample space of 13 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:14 an experiment with n outcomes. Determine the number of events of this experiment. The number of events of this experiment is 2n. 9.393TAN37. Let S be a sample space for an experiment. Show that if E is any event of an experiment, then E and E c are mutually exclusive. If E is an event of an experiment then E c is the event containing the element in S that are not in E. Therefore, E E c and the two sets are mutually exclusive. 14 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:15 In Exercises 40 41 determine whether the statements are true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. 10.393TAN40. If E and F are mutually exclusive and E and G are mutually exclusive, than F and G are mutually exclusive. False. Let E 1, 2, 3, F 4, 5, 6 ,and G 4, 5. Then E F and E G , F G 4, 5 . 11.393TAN41. The numbers 1, 2, and 3 are 15 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:16 written separately on three pieces of paper. These slips of paper are then placed in a bowl. If you draw two slips from the bowl, one at a time, without replacement, then the sample space for this experiment consists of six elements. True. The sample space is S 1, 2, 1, 3, 2, 1, 2, 3, 3, 1, 3, 2. T8.3B9-14: Describe the nature of a sample space for the given experiment, and determine the nature of sample points. 10. Die Roll. Four dice are rolled, and the number that turns up are observed. Solution. 6. 6. 6. 6 6 4 1296. 16 Dr. Raja Latif. Math 131 (043) Sec8.3 Pg:17 12. Ball Selection. From an urn containing eight different balls, four balls are drawn successively without replacement. Solution. 8. 7. 6. 5 1680. 14. Letter Selection. A four-letter "word" is formed by successively choosing any four letters from the alphabet with replacement. Solution. 26. 26. 26. 26 26 4 456976. 17 ... View Full Document