1 Homework (14 pts + 1 bonus) due Jan 21
(1 pt. bonus) Q0. Why were the earrings that I wore to class today relevant to today's lecture The earrings that I wore are dice (clear 6 sided dice to be specific). (3 pts.) 1.4. The following table provides a frequency distribution (in thousands) for the number of rooms in U.S. housing units. Rooms No. Units 1 471 2 1,470 3 11,715 4 23,468 5 24,476 6 21,327 7 13,782 8+ 15,647
If a U.S. housing unit is selected at random, find the probability that it has Total number of units, Total = 471 + 1,470 + 11,715 + 23,468 + 24,476 + 21,327 + 13,782 + 15,647 = 112,356 a) four rooms. No of units with 4 rooms 23,468 = =0.209 total 112,356 b) more than four rooms. No of units with 5 rooms and 6 rooms and 7 rooms and 8 or more rooms total 24,476 + 21,327 + 13,782 + 15,647 75,232 = = =0.670 112,356 112,356 c) one or two rooms No of units with 1 room or 2 rooms 471 + 1,470 1,941 = = = 0.017 total 112,356 112,356 d) fewer than one room. Since all housing units have at least one room, this is the empty set, \ e) one or more rooms. Since all of the units have one or more rooms, this is the whole sample space or 1. f) Identify the population under consideration. The population is the housing units in the U.S.
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(1.5 pt.) 1.10. Use the frequentist interpretation of probability to interpret each statement. a) The probability is 0.314 that the gestation period of a woman will exceed 9 months. Looking at a large number of pregnant women, 314 out of 1000 women will have gestation periods of longer than 9 months. b) The probability is 2/3 that the favorite in a horse race finishes in the money (first, second, or third place). Looking at a large number of horse races, 66 out of 100 favorites will be end up in first, second or third place. c) The probability is 0.40 that a traffic fatality involves an intoxicated or alcohol-impaired driver or nonoccupant. Looking at a large number of traffic accidents that involve fatalities, 40 out of 100 will involve an alcohol impaired driver or an alcohol impaired nonoccupant. (2 pts.) 1.18. Let U = R and, for each n N, define An = [0,1/n]. a) Determine
4 A n-1 n 4 A n-1 n 4 A n-1 n
and
4 A . n-1 n
= [0, 1/1] [0,1/2] [0,1/3] [0,1/4] = [0,1/4] = [0, 1/1] U [0,1/2] U [0,1/3] U [0,1/4] = [0,1]
A n-1 n
b) Determine
A n-1 n
and
A . n-1 n
= [0, 1/1] [0,1/2] [0,1/3] [0,1/4] .... The left side is bounded at 0. The right side gets smaller and smaller but never quite reaches 0. Therefore the answer is {0} (no interval) 4 A = [0, 1/1] U [0,1/2] U [0,1/3] U [0,1/4] U ... n-1 n The left sides is bounded at 0, The right side will be the largest interval which is bounded at 1. Therefore, the answer is [0,1].
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(2 pt.) 1.27(a only). Refer to the distributive as laws, given in Proposition 1.2 on page 12. a) Verify parts (a) and (b) by using Venn diagrams. Proposition 1.2. Distributive Laws Let A, B, and C be subsets of U, Then a) A (B U C) = (A B) U (A C)
A
B
A
B
BUC
C
A (B UC)
C
A
B
A
B
A
B
AB
C
C AC (A B) U (A C)
C
b) A U (B C) = (A U B) (A U C)
A
B
A
B
BC
C
C A U (B C)
A
B
A
B
A
B
AUB
C
C AUC (A U B) (A U C)
C
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(0.5 pt.) 1.15. Draw a Venn diagram showing three subsets, A, B, and C, such that no two are disjoint but that A B C = \. There are many ways that this can be drawn, one way is: A B
C (3 pt.) 2.4 (6-sided die). Suppose that one die is rolled and that you observe the number of dots facing up. a) Obtain a sample space for this random experiment whose elements are integers. The sample space includes all of the possible outcomes or = {1, 2, 3, 4, 5, 6} b) Determine as a subset of the sample space each of the events A = die comes up even, B = die comes up at least 4, C = die comes up at most 2, and D = die comes up 3. A = {2, 4, 6} B = {4, 5, 6} C = {1, 2} D = {3}
c) Determine as a subset of the sample space and describe in words each of the events Ac, A B, and B U C. Ac = {1, 3, 5} which is everything but A OR die comes up odd A B = {4, 6} which is everything that is in A and B OR the die is even and at least 4. B U C = {1, 2, 4, 5, 6} which is the die is NOT 3. f) Describe in words each of the events: {5}, {1, 3, 5}, and {1, 2, 3, 4}. {5}: The event that the die comes up 5. {1, 3, 5}: The event that the die comes up odd. {1, 2, 3, 4}: The event that the die comes up at most 4. (2 pts.) 2.10. This exercise considers two random experiments involving the repeated tossing of a coin. Note: You may assume that eventually a head will be tossed. a) If the coin is tossed until the first time a head appears, find the sample space. = {H, TH, TTH, TTTH, ...} This is the sample space where all the tosses are T except for the last one. b) If the coin is tossed until the second time a head appears, find the sample space. = {HH, THH, HTH, TTHH, THTH, HTTH, ...} This is the sample space where the last toss is a H, and there is only one H in the beginning tosses.
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c) For the experiment in part (a), express the event that the coin is tossed exactly six times in the form {....}, where in place of "...." you list all of the outcomes in that event. E = {TTTTTH} Since the sample space is all tails except for the last toss, this event occurs if you toss 5 tails and then a H. d) Repeat part (c) for the experiment described in part (b). E = {TTTTHH, TTTHTH, TTHTTH, THTTTH, HTTTTH} In this sample space, the last toss is H and there is one other H tossed. Therefore, the first 5 tosses consist of 4 T and 1 H.
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