CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... ·...
Transcript of CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... ·...
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Sample Spaces, Events, Probability
CS 3130/ECE 3530:Probability and Statistics for Engineers
August 28, 2014
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Sets
DefinitionA 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, 31B = apple, pear, orange, grapeNot a valid set definition: C = 1, 2, 3, 4, 2
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Sets
DefinitionA 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, 31B = apple, pear, orange, grapeNot a valid set definition: C = 1, 2, 3, 4, 2
![Page 4: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/4.jpg)
Sets
DefinitionA 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, grapeNot a valid set definition: C = 1, 2, 3, 4, 2
![Page 5: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/5.jpg)
Sets
DefinitionA 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, 31B = apple, pear, orange, grape
Not a valid set definition: C = 1, 2, 3, 4, 2
![Page 6: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/6.jpg)
Sets
DefinitionA 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, 31B = apple, pear, orange, grapeNot a valid set definition: C = 1, 2, 3, 4, 2
![Page 7: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/7.jpg)
SetsI Order in a set does not matter!
1, 2, 3 = 3, 1, 2 = 1, 3, 2
I When x is an element of A, we denote this by:
x ∈ A.
I If x is not in a set A, we denote this as:
x /∈ A.
I The “empty” or “null” set has no elements:
∅ =
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SetsI Order in a set does not matter!
1, 2, 3 = 3, 1, 2 = 1, 3, 2I When x is an element of A, we denote this by:
x ∈ A.
I If x is not in a set A, we denote this as:
x /∈ A.
I The “empty” or “null” set has no elements:
∅ =
![Page 9: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/9.jpg)
SetsI Order in a set does not matter!
1, 2, 3 = 3, 1, 2 = 1, 3, 2I When x is an element of A, we denote this by:
x ∈ A.
I If x is not in a set A, we denote this as:
x /∈ A.
I The “empty” or “null” set has no elements:
∅ =
![Page 10: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/10.jpg)
SetsI Order in a set does not matter!
1, 2, 3 = 3, 1, 2 = 1, 3, 2I When x is an element of A, we denote this by:
x ∈ A.
I If x is not in a set A, we denote this as:
x /∈ A.
I The “empty” or “null” set has no elements:
∅ =
![Page 11: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/11.jpg)
Sample Spaces
DefinitionA sample space is the set of all possible outcomes of anexperiment. We’ll denote a sample space as Ω.
Examples:I Coin flip: Ω = H,TI Roll a 6-sided die: Ω = 1, 2, 3, 4, 5, 6I Pick a ball from a bucket of red/black balls:
Ω = R,B
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Sample Spaces
DefinitionA sample space is the set of all possible outcomes of anexperiment. We’ll denote a sample space as Ω.
Examples:I Coin flip: Ω = H,T
I Roll a 6-sided die: Ω = 1, 2, 3, 4, 5, 6I Pick a ball from a bucket of red/black balls:
Ω = R,B
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Sample Spaces
DefinitionA sample space is the set of all possible outcomes of anexperiment. We’ll denote a sample space as Ω.
Examples:I Coin flip: Ω = H,TI Roll a 6-sided die: Ω = 1, 2, 3, 4, 5, 6
I Pick a ball from a bucket of red/black balls:Ω = R,B
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Sample Spaces
DefinitionA sample space is the set of all possible outcomes of anexperiment. We’ll denote a sample space as Ω.
Examples:I Coin flip: Ω = H,TI Roll a 6-sided die: Ω = 1, 2, 3, 4, 5, 6I Pick a ball from a bucket of red/black balls:
Ω = R,B
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Some Important Sets
I Integers:
Z = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .
I Natural Numbers:
N = 0, 1, 2, 3, . . .
I Real Numbers:
R = “any number that can be written in decimal form”
5 ∈ R, 17.42 ∈ R, π = 3.14159 . . . ∈ R
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Some Important SetsI Integers:
Z = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .
I Natural Numbers:
N = 0, 1, 2, 3, . . .
I Real Numbers:
R = “any number that can be written in decimal form”
5 ∈ R, 17.42 ∈ R, π = 3.14159 . . . ∈ R
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Some Important SetsI Integers:
Z = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .
I Natural Numbers:
N = 0, 1, 2, 3, . . .
I Real Numbers:
R = “any number that can be written in decimal form”
5 ∈ R, 17.42 ∈ R, π = 3.14159 . . . ∈ R
![Page 18: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/18.jpg)
Some Important SetsI Integers:
Z = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .
I Natural Numbers:
N = 0, 1, 2, 3, . . .
I Real Numbers:
R = “any number that can be written in decimal form”
5 ∈ R, 17.42 ∈ R, π = 3.14159 . . . ∈ R
![Page 19: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/19.jpg)
Some Important SetsI Integers:
Z = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .
I Natural Numbers:
N = 0, 1, 2, 3, . . .
I Real Numbers:
R = “any number that can be written in decimal form”
5 ∈ R, 17.42 ∈ R, π = 3.14159 . . . ∈ R
![Page 20: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/20.jpg)
Building Sets Using Conditionals
I Alternate way to define natural numbers:
N = x ∈ Z : x ≥ 0
I Set of even integers:
x ∈ Z : x is divisible by 2
I Rationals:
Q = p/q : p, q ∈ Z, q 6= 0
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Building Sets Using Conditionals
I Alternate way to define natural numbers:
N = x ∈ Z : x ≥ 0
I Set of even integers:
x ∈ Z : x is divisible by 2
I Rationals:
Q = p/q : p, q ∈ Z, q 6= 0
![Page 22: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/22.jpg)
Building Sets Using Conditionals
I Alternate way to define natural numbers:
N = x ∈ Z : x ≥ 0
I Set of even integers:
x ∈ Z : x is divisible by 2
I Rationals:
Q = p/q : p, q ∈ Z, q 6= 0
![Page 23: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/23.jpg)
Building Sets Using Conditionals
I Alternate way to define natural numbers:
N = x ∈ Z : x ≥ 0
I Set of even integers:
x ∈ Z : x is divisible by 2
I Rationals:
Q = p/q : p, q ∈ Z, q 6= 0
![Page 24: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/24.jpg)
Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:
I 1, 9 ⊆ 1, 3, 9, 11I Q ⊆ RI apple, pear * apple, orange, bananaI ∅ ⊆ A for any set AI A ⊆ A for any set A (but A 6⊂ A)
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Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:
I 1, 9 ⊆ 1, 3, 9, 11I Q ⊆ RI apple, pear * apple, orange, bananaI ∅ ⊆ A for any set AI A ⊆ A for any set A (but A 6⊂ A)
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Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:I 1, 9 ⊆ 1, 3, 9, 11
I Q ⊆ RI apple, pear * apple, orange, bananaI ∅ ⊆ A for any set AI A ⊆ A for any set A (but A 6⊂ A)
![Page 27: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/27.jpg)
Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:I 1, 9 ⊆ 1, 3, 9, 11I Q ⊆ R
I apple, pear * apple, orange, bananaI ∅ ⊆ A for any set AI A ⊆ A for any set A (but A 6⊂ A)
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Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:I 1, 9 ⊆ 1, 3, 9, 11I Q ⊆ RI apple, pear * apple, orange, banana
I ∅ ⊆ A for any set AI A ⊆ A for any set A (but A 6⊂ A)
![Page 29: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/29.jpg)
Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:I 1, 9 ⊆ 1, 3, 9, 11I Q ⊆ RI apple, pear * apple, orange, bananaI ∅ ⊆ A for any set A
I A ⊆ A for any set A (but A 6⊂ A)
![Page 30: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/30.jpg)
Subsets
DefinitionA set A is a subset of another set B if every element ofA is also an element of B, and we denote this as A ⊆ B.
Examples:I 1, 9 ⊆ 1, 3, 9, 11I Q ⊆ RI apple, pear * apple, orange, bananaI ∅ ⊆ A for any set AI A ⊆ A for any set A (but A 6⊂ A)
![Page 31: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/31.jpg)
Events
DefinitionAn event is a subset of a sample space.
Examples:
I You roll a die and get an even number:2, 4, 6 ⊆ 1, 2, 3, 4, 5, 6
I You flip a coin and it comes up “heads”:H ⊆ H,T
I Your code takes longer than 5 seconds to run:(5,∞) ⊆ R
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Events
DefinitionAn event is a subset of a sample space.
Examples:
I You roll a die and get an even number:2, 4, 6 ⊆ 1, 2, 3, 4, 5, 6
I You flip a coin and it comes up “heads”:H ⊆ H,T
I Your code takes longer than 5 seconds to run:(5,∞) ⊆ R
![Page 33: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/33.jpg)
Events
DefinitionAn event is a subset of a sample space.
Examples:I You roll a die and get an even number:2, 4, 6 ⊆ 1, 2, 3, 4, 5, 6
I You flip a coin and it comes up “heads”:H ⊆ H,T
I Your code takes longer than 5 seconds to run:(5,∞) ⊆ R
![Page 34: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/34.jpg)
Events
DefinitionAn event is a subset of a sample space.
Examples:I You roll a die and get an even number:2, 4, 6 ⊆ 1, 2, 3, 4, 5, 6
I You flip a coin and it comes up “heads”:H ⊆ H,T
I Your code takes longer than 5 seconds to run:(5,∞) ⊆ R
![Page 35: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/35.jpg)
Events
DefinitionAn event is a subset of a sample space.
Examples:I You roll a die and get an even number:2, 4, 6 ⊆ 1, 2, 3, 4, 5, 6
I You flip a coin and it comes up “heads”:H ⊆ H,T
I Your code takes longer than 5 seconds to run:(5,∞) ⊆ R
![Page 36: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/36.jpg)
Set Operations: Union
DefinitionThe union of two sets A and B, denoted A ∪ B is the setof all elements in either A or B (or both).
When A and B are events, A ∪ B means that event A orevent B happens (or both).
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∪ B = 1, 2, 3, 5
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Set Operations: Union
DefinitionThe union of two sets A and B, denoted A ∪ B is the setof all elements in either A or B (or both).
When A and B are events, A ∪ B means that event A orevent B happens (or both).
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∪ B = 1, 2, 3, 5
![Page 38: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/38.jpg)
Set Operations: Union
DefinitionThe union of two sets A and B, denoted A ∪ B is the setof all elements in either A or B (or both).
When A and B are events, A ∪ B means that event A orevent B happens (or both).
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”
A ∪ B = 1, 2, 3, 5
![Page 39: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/39.jpg)
Set Operations: Union
DefinitionThe union of two sets A and B, denoted A ∪ B is the setof all elements in either A or B (or both).
When A and B are events, A ∪ B means that event A orevent B happens (or both).
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∪ B = 1, 2, 3, 5
![Page 40: CS 3130/ECE 3530: Probability and Statistics for Engineers ...jeffp/teaching/cs3130/... · Probability and Statistics for Engineers August 28, 2014. Sets Definition A set is a collection](https://reader030.fdocuments.us/reader030/viewer/2022040117/5f0b4c147e708231d42fd215/html5/thumbnails/40.jpg)
Set Operations: Intersection
DefinitionThe intersection of two sets A and B, denoted A ∩ B isthe set of all elements in both A and B.
When A and B are events, A ∩ B means that both eventA and event B happen.
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∩ B = 1, 3
Note: If A ∩ B = ∅, we say A and B are disjoint.
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Set Operations: Intersection
DefinitionThe intersection of two sets A and B, denoted A ∩ B isthe set of all elements in both A and B.
When A and B are events, A ∩ B means that both eventA and event B happen.
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∩ B = 1, 3
Note: If A ∩ B = ∅, we say A and B are disjoint.
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Set Operations: Intersection
DefinitionThe intersection of two sets A and B, denoted A ∩ B isthe set of all elements in both A and B.
When A and B are events, A ∩ B means that both eventA and event B happen.
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”
A ∩ B = 1, 3
Note: If A ∩ B = ∅, we say A and B are disjoint.
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Set Operations: Intersection
DefinitionThe intersection of two sets A and B, denoted A ∩ B isthe set of all elements in both A and B.
When A and B are events, A ∩ B means that both eventA and event B happen.
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∩ B = 1, 3
Note: If A ∩ B = ∅, we say A and B are disjoint.
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Set Operations: Intersection
DefinitionThe intersection of two sets A and B, denoted A ∩ B isthe set of all elements in both A and B.
When A and B are events, A ∩ B means that both eventA and event B happen.
Example:A = 1, 3, 5 “an odd roll”B = 1, 2, 3 “a roll of 3 or less”A ∩ B = 1, 3
Note: If A ∩ B = ∅, we say A and B are disjoint.
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Set Operations: Complement
DefinitionThe complement of a set A ⊆ Ω, denoted Ac, is the setof all elements in Ω that are not in A.
When A is an event, Ac means that the event A does nothappen.
Example:A = 1, 3, 5 “an odd roll”Ac = 2, 4, 6 “an even roll”
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Set Operations: Complement
DefinitionThe complement of a set A ⊆ Ω, denoted Ac, is the setof all elements in Ω that are not in A.
When A is an event, Ac means that the event A does nothappen.
Example:A = 1, 3, 5 “an odd roll”Ac = 2, 4, 6 “an even roll”
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Set Operations: Complement
DefinitionThe complement of a set A ⊆ Ω, denoted Ac, is the setof all elements in Ω that are not in A.
When A is an event, Ac means that the event A does nothappen.
Example:A = 1, 3, 5 “an odd roll”
Ac = 2, 4, 6 “an even roll”
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Set Operations: Complement
DefinitionThe complement of a set A ⊆ Ω, denoted Ac, is the setof all elements in Ω that are not in A.
When A is an event, Ac means that the event A does nothappen.
Example:A = 1, 3, 5 “an odd roll”Ac = 2, 4, 6 “an even roll”
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Set Operations: Difference
DefinitionThe difference of a set A ⊆ Ω and a set B ⊆ Ω,denoted A− B, is the set of all elements in Ω that are inA and are not in B.
Example:A = 3, 4, 5, 6B = 3, 5A− B = 4, 6
Note: A− B = A ∩ Bc
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DeMorgan’s Law
Complement of union or intersection:
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
What is the English translation for both sides of theequations above?
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DeMorgan’s Law
Complement of union or intersection:
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
What is the English translation for both sides of theequations above?
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Exercises
Check whether the following statements are true or false.(Hint: you might use Venn diagrams.)
I A− B ⊆ AI (A− B)c = Ac ∪ BI A ∪ B ⊆ BI (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
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Probability
DefinitionA probability function on a finite sample space Ωassigns every event A ⊆ Ω a number in [0, 1], such that
1. P(Ω) = 12. P(A ∪ B) = P(A) + P(B) when A ∩ B = ∅
P(A) is the probability that event A occurs.
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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 equallylikely, then the probability function is given by
P(A) =|A||Ω|
Example: Rolling a 6-sided die
I P(1) = 1/6I P(1, 2, 3) = 1/2
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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 equallylikely, then the probability function is given by
P(A) =|A||Ω|
Example: Rolling a 6-sided die
I P(1) = 1/6I P(1, 2, 3) = 1/2
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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 equallylikely, then the probability function is given by
P(A) =|A||Ω|
Example: Rolling a 6-sided die
I P(1) = 1/6I P(1, 2, 3) = 1/2
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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 equallylikely, then the probability function is given by
P(A) =|A||Ω|
Example: Rolling a 6-sided die
I P(1) = 1/6
I P(1, 2, 3) = 1/2
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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 equallylikely, then the probability function is given by
P(A) =|A||Ω|
Example: Rolling a 6-sided die
I P(1) = 1/6I P(1, 2, 3) = 1/2
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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) 6= (2, 1)Repeats are possible: (1, 1) ∈ N× N
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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) 6= (2, 1)Repeats are possible: (1, 1) ∈ N× N
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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) 6= (2, 1)Repeats are possible: (1, 1) ∈ N× N
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More Repeats
Repeating an experiment n times gives the samplespace
Ωn = Ω× · · · × Ω (n times)
= (x1, x2, . . . , xn) : xi ∈ Ω for all i
The element (x1, x2, . . . , xn) is called an n-tuple.
If |Ω| = k, then |Ωn| = kn.
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More Repeats
Repeating an experiment n times gives the samplespace
Ωn = Ω× · · · × Ω (n times)
= (x1, x2, . . . , xn) : xi ∈ Ω for all i
The element (x1, x2, . . . , xn) is called an n-tuple.
If |Ω| = k, then |Ωn| = kn.
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More Repeats
Repeating an experiment n times gives the samplespace
Ωn = Ω× · · · × Ω (n times)
= (x1, x2, . . . , xn) : xi ∈ Ω for all i
The element (x1, x2, . . . , xn) is called an n-tuple.
If |Ω| = k, then |Ωn| = kn.
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Probability Rules
Complement of an event A:
P(Ac) = 1− P(A)
Union of two overlapping events A ∩ B 6= ∅:
P(A ∪ B) = P(A) + P(B)− P(A ∩ B)
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Probability Rules
Complement of an event A:
P(Ac) = 1− P(A)
Union of two overlapping events A ∩ B 6= ∅:
P(A ∪ B) = P(A) + P(B)− P(A ∩ B)
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Probability Rules
Complement of an event A:
P(Ac) = 1− P(A)
Union of two overlapping events A ∩ B 6= ∅:
P(A ∪ B) = P(A) + P(B)− P(A ∩ B)
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Exercise
You are picking a number out of a hat, which containsthe numbers 1 through 100. What are the followingevents and their probabilities?
I The number has a single digitI The number has two digitsI The number is a multiple of 4I The number is not a multiple of 4I The sum of the number’s digits is 5
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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 isn factorial:
n! = n× (n− 1)× (n− 2)× · · · × 2
How many ways can you rearrange (1, 2, 3, 4)?
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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 isn factorial:
n! = n× (n− 1)× (n− 2)× · · · × 2
How many ways can you rearrange (1, 2, 3, 4)?
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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 isn factorial:
n! = n× (n− 1)× (n− 2)× · · · × 2
How many ways can you rearrange (1, 2, 3, 4)?