6.041 / 6.431 3. Independence of Events

LECTURE 3 • Readings: Section 1.5 • Review • Independence of two events • Independence of a collection of events Review P(A |B)= P(A ! B) P(B) , assuming P(B) > 0 • Multiplication rule: P(A ! B)= P(B) ·P(A |B)= P(A) ·P(B |A) • Total probability theorem: P(B)= P(A)P(B |A)+ P(Ac)P(B |Ac) • Bayes rule: P(Ai |B)= P(Ai)P(B |Ai) P(B) Models based on conditional probabilities • 3 tosses of a biased coin: P(H)= p, P(T )=1 − p p p p p p p p 1 -p 1 -p 1 -p 1 -p 1 -p 1 -p 1 -p HHHHHT HTH HTT THH THT TTH TTT P(T HT )= P(1 head) = P(first toss is H |1 head) = Independence of two events • “Defn:” P(B |A)= P(B) – “occurrence of A provides no information about B’s occurrence” • Recall that P(A ! B)= P(A) ·P(B |A) • Defn: P(A ! B)= P(A) ·P(B) • Symmetric with respect to A and B – applies even if P(A)=0 – implies P(A |B)= P(A) Conditioning may affect independence • Conditional independence, given C, is defined as independence under probability law P( ·|C) • Assume A and B are independent A B C • If we are told that C occurred, are A and B independent? Conditioning may affect independence • Two unfair coins, A and B: P(H | coin A)=0.9, P(H | coin B)=0.1 choose either coin with equal probability 0.9 0.9 0.9 0.1 0.1 0.5 0.5 Coin A Coin B 0.9 0.9 0.9 0.1 0.1 0.1 0.1 • Once we know it is coin A, are tosses independent? • If we do not know which coin it is, are tosses independent? – Compare: P(toss 11 = H) P(toss 11 = H | first 10 tosses are heads) Independence of a collection of events • Intuitive definition: Information on some of the events tells us nothing about probabilities related to the remaining events – E.g.: P(A1 ! (Ac 2 # A3) | A5!Ac 6)= P(A1 ! (Ac 2 # A3)) • Mathematical definition: Events A1,A2, . . . , An are called independent if: P(Ai!Aj!···!Aq)= P(Ai)P(Aj) ···P(Aq) for any distinct indices i, j, . . . , q, (chosen from {1, . . . , n}) Independence vs. pairwise independence • Two independent fair coin tosses – A: First toss is H – B: Second toss is H – P(A)= P(B)=1/2 HH HT TH TT – C: First and second toss give same result – P(C)= – P(C ! A)= – P(A ! B ! C)= – P(C | A ! B)= • Pairwise independence does not imply independence The king’s sibling • The king comes from a family of two children. What is the probability that his sibling is female? MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.