# Sample Spaces, Events, Probability

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1 Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014

2 Sets A set is a collection of unique objects.

3 Sets A set is a collection of unique objects. Here objects can be concrete things (people in class, schools in PAC-12), or abstract things (numbers, colors).

4 Sets A set is a collection of unique objects. Here objects can be concrete things (people in class, schools in PAC-12), or abstract things (numbers, colors). Examples: A = {3, 8, 31}

5 Sets A set is a collection of unique objects. Here objects can be concrete things (people in class, schools in PAC-12), or abstract things (numbers, colors). Examples: A = {3, 8, 31} B = {apple, pear, orange, grape}

6 Sets A set is a collection of unique objects. Here objects can be concrete things (people in class, schools in PAC-12), or abstract things (numbers, colors). Examples: A = {3, 8, 31} B = {apple, pear, orange, grape} Not a valid set definition: C = {1, 2, 3, 4, 2}

7 Sets Order in a set does not matter! {1, 2, 3} = {3, 1, 2} = {1, 3, 2}

8 Sets Order in a set does not matter! {1, 2, 3} = {3, 1, 2} = {1, 3, 2} When x is an element of A, we denote this by: x A.

9 Sets Order in a set does not matter! {1, 2, 3} = {3, 1, 2} = {1, 3, 2} When x is an element of A, we denote this by: x A. If x is not in a set A, we denote this as: x / A.

10 Sets Order in a set does not matter! {1, 2, 3} = {3, 1, 2} = {1, 3, 2} When x is an element of A, we denote this by: x A. If x is not in a set A, we denote this as: x / A. The empty or null set has no elements: = { }

11 Sample Spaces A sample space is the set of all possible outcomes of an experiment. We ll denote a sample space as Ω.

12 Sample Spaces A sample space is the set of all possible outcomes of an experiment. We ll denote a sample space as Ω. Examples: Coin flip: Ω = {H, T}

13 Sample Spaces A sample space is the set of all possible outcomes of an experiment. We ll denote a sample space as Ω. Examples: Coin flip: Ω = {H, T} Roll a 6-sided die: Ω = {1, 2, 3, 4, 5, 6}

14 Sample Spaces A sample space is the set of all possible outcomes of an experiment. We ll denote a sample space as Ω. Examples: Coin flip: Ω = {H, T} Roll a 6-sided die: Ω = {1, 2, 3, 4, 5, 6} Pick a ball from a bucket of red/black balls: Ω = {R, B}

15 Some Important Sets

16 Some Important Sets Integers: Z = {..., 3, 2, 1, 0, 1, 2, 3,...}

17 Some Important Sets Integers: Z = {..., 3, 2, 1, 0, 1, 2, 3,...} Natural Numbers: N = {0, 1, 2, 3,...}

18 Some Important Sets Integers: Z = {..., 3, 2, 1, 0, 1, 2, 3,...} Natural Numbers: N = {0, 1, 2, 3,...} Real Numbers: R = any number that can be written in decimal form

19 Some Important Sets Integers: Z = {..., 3, 2, 1, 0, 1, 2, 3,...} Natural Numbers: N = {0, 1, 2, 3,...} Real Numbers: R = any number that can be written in decimal form 5 R, R, π = R

20 Building Sets Using Conditionals

21 Building Sets Using Conditionals Alternate way to define natural numbers: N = {x Z : x 0}

22 Building Sets Using Conditionals Alternate way to define natural numbers: N = {x Z : x 0} Set of even integers: {x Z : x is divisible by 2}

23 Building Sets Using Conditionals Alternate way to define natural numbers: N = {x Z : x 0} Set of even integers: {x Z : x is divisible by 2} Rationals: Q = { p/q : p, q Z, q 0}

24 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B.

25 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B. Examples:

26 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B. Examples: {1, 9} {1, 3, 9, 11}

27 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B. Examples: {1, 9} {1, 3, 9, 11} Q R

28 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B. Examples: {1, 9} {1, 3, 9, 11} Q R {apple, pear} {apple, orange, banana}

29 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B. Examples: {1, 9} {1, 3, 9, 11} Q R {apple, pear} {apple, orange, banana} A for any set A

30 Subsets A set A is a subset of another set B if every element of A is also an element of B, and we denote this as A B. Examples: {1, 9} {1, 3, 9, 11} Q R {apple, pear} {apple, orange, banana} A for any set A A A for any set A (but A A)

31 Events An event is a subset of a sample space.

32 Events An event is a subset of a sample space. Examples:

33 Events An event is a subset of a sample space. Examples: You roll a die and get an even number: {2, 4, 6} {1, 2, 3, 4, 5, 6}

34 Events An event is a subset of a sample space. Examples: You roll a die and get an even number: {2, 4, 6} {1, 2, 3, 4, 5, 6} You flip a coin and it comes up heads : {H} {H, T}

35 Events An event is a subset of a sample space. Examples: You roll a die and get an even number: {2, 4, 6} {1, 2, 3, 4, 5, 6} You flip a coin and it comes up heads : {H} {H, T} Your code takes longer than 5 seconds to run: (5, ) R

36 Set Operations: Union The union of two sets A and B, denoted A B is the set of all elements in either A or B (or both).

37 Set Operations: Union The union of two sets A and B, denoted A B is the set of all elements in either A or B (or both). When A and B are events, A B means that event A or event B happens (or both).

38 Set Operations: Union The union of two sets A and B, denoted A B is the set of all elements in either A or B (or both). When A and B are events, A B means that event A or event B happens (or both). Example: A = {1, 3, 5} B = {1, 2, 3} an odd roll a roll of 3 or less

39 Set Operations: Union The union of two sets A and B, denoted A B is the set of all elements in either A or B (or both). When A and B are events, A B means that event A or event B happens (or both). Example: A = {1, 3, 5} B = {1, 2, 3} A B = {1, 2, 3, 5} an odd roll a roll of 3 or less

40 Set Operations: Intersection The intersection of two sets A and B, denoted A B is the set of all elements in both A and B.

41 Set Operations: Intersection The intersection of two sets A and B, denoted A B is the set of all elements in both A and B. When A and B are events, A B means that both event A and event B happen.

42 Set Operations: Intersection The intersection of two sets A and B, denoted A B is the set of all elements in both A and B. When A and B are events, A B means that both event A and event B happen. Example: A = {1, 3, 5} B = {1, 2, 3} an odd roll a roll of 3 or less

43 Set Operations: Intersection The intersection of two sets A and B, denoted A B is the set of all elements in both A and B. When A and B are events, A B means that both event A and event B happen. Example: A = {1, 3, 5} B = {1, 2, 3} A B = {1, 3} an odd roll a roll of 3 or less

44 Set Operations: Intersection The intersection of two sets A and B, denoted A B is the set of all elements in both A and B. When A and B are events, A B means that both event A and event B happen. Example: A = {1, 3, 5} B = {1, 2, 3} A B = {1, 3} an odd roll a roll of 3 or less Note: If A B =, we say A and B are disjoint.

45 Set Operations: Complement The complement of a set A Ω, denoted A c, is the set of all elements in Ω that are not in A.

46 Set Operations: Complement The complement of a set A Ω, denoted A c, is the set of all elements in Ω that are not in A. When A is an event, A c means that the event A does not happen.

47 Set Operations: Complement The complement of a set A Ω, denoted A c, is the set of all elements in Ω that are not in A. When A is an event, A c means that the event A does not happen. Example: A = {1, 3, 5} an odd roll

48 Set Operations: Complement The complement of a set A Ω, denoted A c, is the set of all elements in Ω that are not in A. When A is an event, A c means that the event A does not happen. Example: A = {1, 3, 5} A c = {2, 4, 6} an odd roll an even roll

49 Set Operations: Difference The difference of a set A Ω and a set B Ω, denoted A B, is the set of all elements in Ω that are in A and are not in B. Example: A = {3, 4, 5, 6} B = {3, 5} A B = {4, 6} Note: A B = A B c

50 DeMorgan s Law Complement of union or intersection: (A B) c = A c B c (A B) c = A c B c

51 DeMorgan s Law Complement of union or intersection: (A B) c = A c B c (A B) c = A c B c What is the English translation for both sides of the equations above?

52 Exercises Check whether the following statements are true or false. (Hint: you might use Venn diagrams.) A B A (A B) c = A c B A B B (A B) C = (A C) (B C)

53 Probability A probability function on a finite sample space Ω assigns every event A Ω a number in [0, 1], such that 1. P(Ω) = 1 2. P(A B) = P(A) + P(B) when A B = P(A) is the probability that event A occurs.

54 Equally Likely Outcomes The number of elements in a set A is denoted A.

55 Equally Likely Outcomes The number of elements in a set A is denoted A. If Ω has a finite number of elements, and each is equally likely, then the probability function is given by P(A) = A Ω

56 Equally Likely Outcomes The number of elements in a set A is denoted A. If Ω has a finite number of elements, and each is equally likely, then the probability function is given by P(A) = A Ω Example: Rolling a 6-sided die

57 Equally Likely Outcomes The number of elements in a set A is denoted A. If Ω has a finite number of elements, and each is equally likely, then the probability function is given by P(A) = A Ω Example: Rolling a 6-sided die P({1}) = 1/6

58 Equally Likely Outcomes The number of elements in a set A is denoted A. If Ω has a finite number of elements, and each is equally likely, then the probability function is given by P(A) = A Ω Example: Rolling a 6-sided die P({1}) = 1/6 P({1, 2, 3}) = 1/2

59 Repeated Experiments If we do two runs of an experiment with sample space Ω, then we get a new experiment with sample space Ω Ω = {(x, y) : x Ω, y Ω}

60 Repeated Experiments If we do two runs of an experiment with sample space Ω, then we get a new experiment with sample space Ω Ω = {(x, y) : x Ω, y Ω} The element (x, y) Ω Ω is called an ordered pair.

61 Repeated Experiments If we do two runs of an experiment with sample space Ω, then we get a new experiment with sample space Ω Ω = {(x, y) : x Ω, y Ω} The element (x, y) Ω Ω is called an ordered pair. Properties: Order matters: (1, 2) (2, 1) Repeats are possible: (1, 1) N N

62 More Repeats Repeating an experiment n times gives the sample space Ω n = Ω Ω (n times) = {(x 1, x 2,..., x n ) : x i Ω for all i}

63 More Repeats Repeating an experiment n times gives the sample space Ω n = Ω Ω (n times) = {(x 1, x 2,..., x n ) : x i Ω for all i} The element (x 1, x 2,..., x n ) is called an n-tuple.

64 More Repeats Repeating an experiment n times gives the sample space Ω n = Ω Ω (n times) = {(x 1, x 2,..., x n ) : x i Ω for all i} The element (x 1, x 2,..., x n ) is called an n-tuple. If Ω = k, then Ω n = k n.

65 Probability Rules

66 Probability Rules Complement of an event A: P(A c ) = 1 P(A)

67 Probability Rules Complement of an event A: P(A c ) = 1 P(A) Union of two overlapping events A B : P(A B) = P(A) + P(B) P(A B)

68 Exercise You are picking a number out of a hat, which contains the numbers 1 through 100. What are the following events and their probabilities? The number has a single digit The number has two digits The number is a multiple of 4 The number is not a multiple of 4 The sum of the number s digits is 5

69 Permutations A permutation is an ordering of an n-tuple. For instance, the n-tuple (1, 2, 3) has the following permutations: (1, 2, 3), (1, 3, 2), (2, 1, 3) (2, 3, 1), (3, 1, 2), (3, 2, 1)

70 Permutations A permutation is an ordering of an n-tuple. For instance, the n-tuple (1, 2, 3) has the following permutations: (1, 2, 3), (1, 3, 2), (2, 1, 3) (2, 3, 1), (3, 1, 2), (3, 2, 1) The number of unique orderings of an n-tuple is n factorial: n! = n (n 1) (n 2) 2

71 Permutations A permutation is an ordering of an n-tuple. For instance, the n-tuple (1, 2, 3) has the following permutations: (1, 2, 3), (1, 3, 2), (2, 1, 3) (2, 3, 1), (3, 1, 2), (3, 2, 1) The number of unique orderings of an n-tuple is n factorial: n! = n (n 1) (n 2) 2 How many ways can you rearrange (1, 2, 3, 4)?

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