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Math 76O-143 Finite Mathematics
Self-marking homework assignment #6x
EARLY TOPICS IN PROBABILITY (TERMINOLGY AND COUNTING)
Submit over the internet by 11:59 PM on Sunday 17 April 2016
Maximum value toward semester grade is 2 pts
Methods will be discussed in class;
CAUTION: Prof McFarland makes new tests each semester.
Average grade on Fall 2015 paper test #6 : 27.66 Perfect = 45 See future grade prospects?

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The following 10 questions have equal value, scored one point apiece.
To earn that one point, an answer must be totally correct (no partial credit awarded)

Review of terminology and set notation used in this self-test
Sample Space S : the set of outcomes of an experiment. (See very simple examples)
Uniform Sample Space : all outcomes are equally likely
Event : A subset of a sample space S.

  1. If S is a sample space containing 3 outcomes A1, A2, and A3, then we write this statement S = {A1,A2,A3}
  2. If S = {A1,A2,A3} is a sample space, then A1, A2, and A3 are outcomes in S, but {A1,A2,A3} is not an outcome. A set cannot be a member of itself.
  3. If S = {A1,A2,A3} as above, then {A1} is a subset of S, and is also the event in which the outcome A1 has occured. Events and sample spaces are written using set brackets { }; the outcomes inside them are written without these brackets.
  4. If S = {A1,A2,A3} as above, then E = {A1, A2} is the event or subset of S in which either outcome A1 or outcome A2 has occured (they cannot both occur). Again note that {A1, A2} is not an outcome of S, but A1 and A2 are outcomes inside of both E and S

 

Review of the way probabilities are calculated (for finite uniform sample spaces)
  1. Identify the experiment. What, exactly, is being done.
  2. Identify the Sample Space S. Write out a few outcomes of the experiment (as in problems [1] to [10] below)
  3. Noting how S is built, count the size of S using the appropriate tool, that is, how many different outcomes does the experiment have (as in [3], [6], [8], and [10] below)?
  4. Identify the event E. Write down a few outcomes in E (as in [2], [5], [7], and [9] below)
  5. Noting how E is built, count the size of E using the appropriate tool, that is, in how many different ways can E occur (as in [3], [6], [8], and [10] below)?
  6. The probability of E is then [ the size of E ] divided by [ the size of S ], that is, the probability of E is then
    [ the number of ways the event can occur ] divided by [ the number of ways the experiment can end ],
An example of the above procedure (similar to examples done in class):

Question: What is the probability that a 5-card poker hand contains exactly 2 aces?
The experiment: Draw 5 cards (that is, a quintuple) from a 52-card deck
The sample space S: the set of quintuples, such as { K♠, K♣, 4♥, J♥, 3♦ }.
The size of S: C(52,5) since a poker hand is an unordered 5-card subset of the deck
The event E: the set of quintuples which include exactly 2 aces, such as { Q♠, A♣, A♥, 2♦, 3♦ }
The size of E: C(4,2) aces, C(48,3) non-aces, so size of E is the product C(4,2)C(48,3) using multiplication rule
The probability of E: [C(4,2)C(48,3)] divided by C(52.5), which is about 0.04

[1]
On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" X consists of randomly choosing one of these kids as RA.
Which of the following statements are true for the Sample Space of experiment X?
(a) A1 is an outcome in the Sample Space
(b) A2 is an outcome in the Sample Space
(c) A3 is an outcome in the Sample Space
(d) A3, A4, and A7 are 3 outcomes in the Sample Space
(e) {A3, A4, and A7} is an outcome in the Sample Space
(f) {A2} is an outcome in the Sample Space
(g) {A3} is an outcome in the Sample Space

[2]
(Continuing #1 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" X consists of randomly choosing one of these kids as RA. E is the event that A2 is chosen to be RA.
(b)(1 pt) Which of the following are true statements about the event E?
(a) A1 is an outcome in the event E
(b) A2 is an outcome in the event E
(c) A3 is an outcome in the event E
(d) A1, A2, and A7 are 3 outcomes in the event E
(e) {A1, A2, A3} is an outcome in the event E
(f) {A2} is an outcome in the event E
(g) {A3} is an outcome in the event E

[3]
(Continuing #2 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" X consists of randomly choosing one of these kids as RA. E is the event that A2 is chosen to be RA.
(a) What is the size of the Sample Space for the experiment X?
(b) What is the size of the event E?
(c) What is the probability of the event E?

[4]
On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee.
Which of the following statements are true for the Sample Space of experiment Y?
(a) A1 is an outcome in the Sample Space
(b) A2 is an outcome in the Sample Space
(c) A3 is an outcome in the Sample Space
(d) A1, A2, and A3 are 3 outcomes in the Sample Space
(e) {A1, A2, A3 } is an outcome in the Sample Space
(f) A1, A3, and A5 are 3 outcomes in the Sample Space
(g) {A1, A3, A5 } is an outcome in the Sample Space

[5]
(Continuing #4 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee. F (in English) is the event that the chosen kids are A1, A2, and A3
Which of the following are true statements about the event F?
(a) A1 is an outcome in the event F
(b) A2 is an outcome in the event F
(c) A3 is an outcome in the event F
(d) A1, A2, and A3 are 3 outcomes in the event F
(e) {A1, A2, A3 } is an outcome in the event F
(f) A1, A3, and A5 are 3 outcomes in the event F
(g) {A1, A3, A5 } is an outcome in the event F

[6]
(Continuing #5 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee. F (in English) is the event that the chosen kids are A1, A2, and A3
(a) What is the size of the Sample Space for the experiment Y?
(b) What is the size of the event F?
(c) What is the probability of the event F?
Review counting techniques?

[7]
(Continuing #4 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee. G (in English) is the event that NONE of the kids A6, A7, and A8 are chosen to be on the clean-up committee.
Which of the following are true statements about the event G?
(a) A1 is an outcome in the event G
(b) A2 is an outcome in the event G
(c) A3 is an outcome in the event G
(d) A1, A2, and A3 are 3 outcomes in the event G
(e) {A1, A2, A3 } is an outcome in the event G
(f) A1, A2, and A6 are 3 outcomes in the event G
(g) {A1, A2, A6 } is an outcome in the event G

[8]
(Continuing #7 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee. G (in English) is the event that NONE of the kids A6, A7, and A8 are chosen to be on the clean-up committee.
(a) What is the size of the Sample Space for the experiment Y?
(b) What is the size of the event G?
(c) What is the probability of the event G?
Review counting techniques?

[9]
(Continuing #4 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee. H (in English) is the event that A1 is chosen to be on the clean-up committee.
Which of the following are true statements about the event H?
(a) A1 is an outcome in the event H
(b) A2 is an outcome in the event H
(c) {A1 } is an outcome in the event H
(d) A1, A2, and A3 are 3 outcomes in the event H
(e) {A1, A2, A3 } is an outcome in the event H
(f) A1, A2, and A6 are 3 outcomes in the event H
(g) {A1, A2, A6 } is an outcome in the event H

[10]
(Continuing #9 above) On your dorm floor live 8 kids named A1, A2, A3, A4, A5, A6, A7, and A8. An "experiment" Y consists of randomly choosing three of these kids as a clean-up committee. H (in English) is the event that A1 is chosen to be on the clean-up committee.
(a) What is the size of the Sample Space for the experiment Y?
(b) What is the size of the event H?
(c) What is the probability of the event H?
Review counting techniques?

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