Discrete Probability

Introduction to Discrete Probability

• Key Terms
  • experiment
    • a procedure that yields one of a given set of possible outcomes
    • Ex) rolling a dice
  • sample space
    • the set of possible outcomes
    • Ex) {1, 2, 3, 4, 5 ,6}
  • event
    • a subset of the sample space
    • Ex) {2, 4, 6}
• Probability of an Event
  • S: a finite sample space of equally likely outcomes
  • E: an event
  • Probability of E: P(E) = |E| / |S|
  • 0≤ 𝑝(𝐸) ≤1
• Complements
  • 𝑝(𝐸-bar)=1−𝑝(𝐸)
• Unions
  • 𝑝(𝐸1∪𝐸2) =𝑝(𝐸1) +𝑝(𝐸2) − 𝑝(𝐸1 ∩ 𝐸2)

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Probability Theory

• Probability distribution (확률분포)
  • the function p from the set of all outcomes of the sample space S
  • 0≤ 𝑝(𝑠) ≤1 for each 𝑠∈𝑆
  • ∑𝑠∈𝑆 𝑝(𝑠) = 1
• Uniform Distribution (항등분포)
  • 집합 S의 각 element의 확률 = 1/n
  • p(1) = p(2) = ∙∙∙ = p(n) = 1/n
• Probability of an Event
  • p(E) = ∑𝑠∈𝑆 𝑝(𝑠)
• Complements and Unions
  • 𝑝(𝐸-bar)=1−𝑝(𝐸)
  • ∑s∈S 𝑝(𝑠) = 1 = 𝑝(𝐸) + 𝑝(𝐸-bar)
  • 𝑝(𝐸1∪𝐸2) =𝑝(𝐸1) +𝑝(𝐸2) − 𝑝(𝐸1 ∩ 𝐸2)
• Combinations of Events
  • E1, E2, … is a sequence of pairwise disjoint events in a sample space S
• Conditional Probability (조건부 확률)
  • p(F) > 0
  • p(E|F) = 𝑝(𝐸∩𝐹)/p(F)
• Independence (독립)
  • 𝑝(𝐸∩𝐹) = p(E)p(F)
• Pairwise and Mutual Independence
  • pairwise independent
    • P(A∩B) = P(A)P(B)
    • P(A∩C) = P(A)P(C)
    • P(B∩C) = P(B)P(C)
  • mutually independent
    • P(A∩B∩C) = P(A)P(B)P(C)
    • P(A∩B) = P(A)P(B)
    • P(A∩C) = P(A)P(C)
    • P(B∩C) = P(B)P(C)
• Bernoulli Trials
  • only two possible outcomes(success and failure)
  • p: probability of success
  • q: probability of failure
  • p+q=1
• Binomial distribution
  • b(k:n, p) = C(n,k)p^k*q^(n-k)