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)
Exercise 2. The Principle of Inclusion-Exclusion Show that P(A∪B) = P(A) + P(B)−P(A∩B) (1.4) By writing...
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?
2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)
Read the following problem. Use your knowledge about the Inclusion-Exclusion Principle to support your criteria. Telephone numbering is an application of the inclusion-exclusion principle. Discuss with your peers a way in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry). For each new numbering plan, you discuss show how to find the number of different telephone numbers...
Read the following problem. Use your knowledge about the Inclusion-Exclusion Principle to support your criteria. Telephone numbering is an application of the inclusion-exclusion principle. Discuss with your peers a way in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry). For each new numbering plan you discuss show how to find the number of different telephone numbers...
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
Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle.
Please do only Problem 4! Use 3 as result. 3. Use the inclusion-exclusion formula derived in class as well as induction on the integer n to show that for any sequence of events {AjlI, we have that j-1 This upper bound is referred to as the union bound. 4. Extend the above result to show that we have the analogous bound P( A) P(A), j-1 for the case of an arbitrary, but countable, number of events } Hint: Use the...
Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can be written as a disjoint union in the form A ∪ B = (A \ (A ∩ B)) ∪˙ (B \ (A ∩ B)) ∪˙ (A ∩ B). Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle. Exercise 1.10. Prove for...
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair die 10 times. Call a number in 1, 2, 3, 4, 5, 6 a loner if it is rolled exactly once on the 10 rolls. (For example, if the rolls are 1 2 6 4 4 4 6 3 4 1, then 2 and 3 are the only loners) Compute the probability that at least one of numbers 1, 2, 3 is a loner....
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.