Conditional Probability Definition: The conditional probability that event E occurs given that event &2 occurs is Pr8|E2]= Pr[E1AE2] Pr(E2] For independent E1,E2, Pr[E1|e2]= Pr[E1∧E2] Pr[E1 E2] Pr(E2] Pr[e]·Pr[e2] 2 Pr[E2] =Pr[E1]
Conditional Probability E1 E2 Pr[E1 | E2] Definition: The conditional probability that event E1 occurs given that event E2 occurs is Pr[E1 | E2] = Pr[E1⇥E2] Pr[E2] . For independent E1, E2, Pr[E1 | E2] = Pr[E1 ⇤ E2] Pr[E2] = Pr[E1] · Pr[E2] Pr[E2] = Pr[ E1]
Law of Total Probability Law of total probability: m For disjoint E1,82,...,n that 8=2, m m 2 Pr[E]=Pr[eAE=>Pr[EE]Pr[Ei. i=1 i=1 Analyze the probability by cases!
Law of Total Probability Law of total probability: Pr[E] = n i=1 Pr[E ⇤ Ei] = n i=1 Pr[E|Ei] · Pr[Ei]. For disjoint E1, E2,..., En that n i Ei = , Analyze the probability by cases!
Law of Successive Conditioning (chain rule) Theorem For any E1,E2,...,n, 区- i<飞 Proof: m Pr E m-1 Pr Li=1 m-1 51 Pr recursion! 2=1
Law of Successive Conditioning For any E1, E2,..., En, Pr ⌅ n i=1 Ei ⇥ = ⇤ n k=1 Pr Ek | ⌅ i<k Ei ⇥ . Theorem Proof: recursion! Pr En n 1 i=1 Ei Pr n i=1 Ei Pr n 1 i=1 Ei = (chain rule)
Random Variables probability space: X is the outcome (2,∑，Pr) 1 2 random variable X 3 4 5 6
Random Variables random variable X X is the outcome 1 2 3 4 5 6 (, ,Pr) probability space:
Random Variables probability space: X indicates the evenness (2,∑，Pr) random variable X a function defined over the sample space X:2→R
Random Variables random variable X a function defined over the sample space X : R (, ,Pr) probability space: X indicates the evenness 0 1