here P(A U B U C) =P((A u B) u C) =P(A u B)+P(C)-P((A u B) n C)
=P(A)+P(B)-P(A n B)+P(C)-P((A+B - A n B) n C)
=P(A)+P(B)+P(C)-P(A n B)-(P(A n C)+P(B n C)-P(A n B n C))
P(A U B U C)=P(A)+P(B)+P(C)-P(A n B)-P(A n C)-P(B n C)+P(A n B n C)
2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)
Exercise 4. By writing AU BUC as (AUB) UC, show that the Principle of Inclusion-Exclusion for three sets is P(AUBUC) = P(A)+P(B)+P(C)- P(ANB) - P(ANC) - P(BNC)+P(ANBNC) Can you generalize the result to an arbitrary number of events?
Exercise 2. The Principle of Inclusion-Exclusion Show that P(A∪B) = P(A) + P(B)−P(A∩B) (1.4) By writing A∪B∪C as (A∪B)∪C, extend the result to three sets: P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)
5. In each step, explain clearly what property or axiom you are using (a) Prove "inclusion-exclusion," that P(AUB) P(A) P(B) P(AnB). (b) Prove the "union bound," that P(Ai UA2) P(A) +P(A2). Under what conditions does the equality hold?
5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion," that P(AUB) P(A) +P(B)-P(AnB). (b) Prove the "union bound," that P(AiUAz) < P(A) + P(A2). Under what conditions does the equality hold? (c) Prove that, for Ai and A2 disjoint, P(Ai UA2|B) P(AiB)P(A2|B) (d) A and B are independent events with nonzero probability. Prove whether or not A and B are independent.
5. In each step. explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion," that P(AUB) P(A) +P(B)-PAnB). (b) Prove the "uni that P(A UA2) S P(Ai)+ P(A2). Under what conditions does the on bound." equality hold? (c) Prove that, for A1 and A2 disjoint, PAUA2lB)=P(A1B)+P(A21B) (d) A and B are independent events with nonzero probability. Prove whether or not A and B are independent.
Exercise 9. (Submit a) a) Prove (p4) directly using inclusion-exclusion Hint: With n= pi p?... per set A; = {me [n] |P: | m} Then E (n) = N A
3. Using only the three axioms of probability, prove the Bonferroni inequality: P(AUB P(A) P(B)
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat, for A1 andA2 disjoint, P(A1∪A2|B)=P(A1|B)+P(A2|B). 4. A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.
5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,'' that PAU B) = P(A)-P(B)-Pan B). (b) Prove the "union bound" that P(Ai UA2) P(Ai) P(A2). Under what conditions does the equality hold? (c) Prove that, for A1 and A-disjoint, PAUA2B)= PAB)-P(A-B). d) A and B are independent events with nonzero probability. Prove whether or not A and B are independent.
using the principle of inclusion-exclusion, find the number of solutions of the equation u1+u2+...+u6 = 15, where ui<6,i=1,..,6.