Chapter 1. Experiments and Probabilities Doug Young Suh Media Lab.

1.1 Set and Probability MediaLab, Kyunghee University2 Universal set S = {1, 2, 3, 4, 5, 6} Element 1, 2, 3, 4, 5, 6 Subset A = {even} = {2, 4, 6} B = {odd} = {1, 3, 5} 2, 4, 6 A 1, 3, 5 B

1.1 Set Theory Mathematical basis of probability  set theory Basic definitions (set = event, element = outcome) ◈ Set : a collection of things (denoted by A, B, C) ◈ Elements : the things that make up the set (denoted by x, y, z) Inclusion ( ∈ ) x ∈ A : x is an element of set A. 범하기 쉬운 오류 {x} ∈ A x ∉ A : x is not an element of set A. Subset ( ⊂ ) A ⊂ B: The set A is the subset of set B. Set equality (=) A = B : if and only if A ⊂ B and B ⊂ A MediaLab, Kyunghee University3

Two important sets Universal set (S) Ex) A possible universal set of A : S = { 0, 1, …} Then, A ⊂ S Ex) dice S = {1, 2, 3, 4, 5, 6} Null set (Φ) : The set with no element Note) For any set A, Φ ⊂ A MediaLab, Kyunghee University4

Set algebra Union of two sets A and B : A ∪ B x ∈ A ∪ B if and only if x ∈ A or x ∈ B Intersection of two sets A and B : A ∩ B x ∈ A ∩ B if and only if x ∈ A and x ∈ B Complement of a set A : A c x ∈ A c if and only if x ∉ A Difference between A and B : A - B x ∈ A - B if and only if x ∈ A and x ∉ B Note) Difference is a combination of intersection and complement A – B = A ∩ B c MediaLab, Kyunghee University5 A B

Mutually exclusive and collectively exhaustive Mutually exclusive A collection of sets A 1, A 2,…,A n are mutually exclusive if and only if A i ∩A j = Φ, i ≠ j Note) If there are only two sets in the collection, the sets are called "disjoint". Collectively exhaustive A collection of sets A 1, A 2,…,A n are collectively exhaustive if and only if A 1 ∪ A 2 ∪ … ∪ A n = S MediaLab, Kyunghee University6 1,2 3,4 5,6 A1A1 A 2 A 3

MediaLab, Kyunghee University7 A B S

1.2 Probability Axioms MediaLab, Kyunghee University8 Definition 1.4 Theorem 1.1 Theorem 1.2

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1.2 Probability Axioms MediaLab, Kyunghee University10 2, 4, 6 A 1, 3, 5 B Theorem 1.4 Theorem 1.5

1.3 Conditional Probability Applications: Pattern recognition (signal  information), man-machine interface, pervasive computer, data compression etc. Toss twice, first for X, second for Y. Z=X+Y Event A = {X|3<X<6} Event B = {Y|2<X<5} Event C = {Z|Z>4} Conditional probability P[A|B] "the probability of A given B" MediaLab, Kyunghee University

1.3 Conditional Probability MediaLab, Kyunghee University12 P(A|C) P(B|C) P(C|A) A B C Definition 1.5 Theorem 1.6

1.4 Partitions and total probability A MediaLab, Kyunghee University13

1.4 Partitions and total probability MediaLab, Kyunghee University14

1.4 Partitions and total probability MediaLab, Kyunghee University15 Theorem 1.9 Theorem 1.10

1.5 Independence Disjoint P[AB] = 0 MediaLab, Kyunghee University A B AB Definition 1.6

1.6 Independence Disjoint P[AB] = 0 MediaLab, Kyunghee University17 Definition 1.7 Definition 1.8

problems ~ ~ ~13 MediaLab, Kyunghee University18