For any events A, B, C, and D = A∩B∩C prove the following equality: P(D^c) = 1−P(A)·P(B | A)·P(C | A∩B)
For any events A, B, C, and D = A∩B∩C prove the following equality: P(D^c) =...
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.
Prove/ Disprove : If A and B are any two events, then it is always true that P (A ∪ B) ≤ P (A) + P (B)
3. Let A and B be events. Show that P(ABl(AUB)) P(AB|A). When does equality hold?
3. Let A and B be events. Show that P(AB(AUB)) P(ABA). When does equality hold?
3. Let A, B, C be events in a sample space S. Prove that (a) P(AUB) P(A)P(B), (b) P(AUBUC) P(A)+P(B)+P(C)-P(AnB)-P(Anc)-P(Bnc)+P(AnBnc)
Given the following information about events A, B. and C, determine which pairs of events,if any, are independent and which pairs are mutually exclusive. P(A)-0.3 P(BIA) 0.3 P(B)0.5 P(CB) 0.33 P(C) 0.33 P(AIC)-0.33 Select all correct answers. Select all that apply: A and Care mutually exclusive D A and Care independent O Band C are independent 0 Band C are mutually exclusive D Aand B are mutualy exclusive A and B are independ
Question 4 You are given the following information on Events A, B, C, and D. P(A) = .5 P(B) = .3 P (C) = .15 P(A U D) = .7 P(A ∩ C) = 0.05 P (A │B) = 0.22 P (A ∩ D) = 0.25 Compute P(D). Compute P(A ∩ B). Compute P(A | C). Compute the probability of the complement of C. What does it mean to be mutually exclusive? Give an example of two events that are...
Given the following: A, B, and C are events. P[A] = 0.3 P[B] = 0.3 P[C] = 0.55 P[A intersect B] = 0 P[A' intersect B' intersect C'] = 0.1 P[A intersect C'] = 0.2 (i) Write a set expression for each of the following events a through d. (ii) Find the probability of the event. (Please show all work. Use venn diagrams if necessary). (a) At least one of the events A, B, or C occurs. (b) Exactly one...
An are any n1 events I. Prove the following theorem by mathematical induction: If Ao, A, such that P(AoA,A2 A 0, ten
1. Consider dependent events C and D. P(C and D) = 0.018, P(C) = 0.3, P(D) = 0.5 Find p(C|D) and P(C|D) 2. Consider dependent events E and F P(E and F) = 0.072, P(E) = 0.06, P(F) = 0.09. Find P(F|E) 3. Consider dependent events A and B. P(A and B) = 0.036, P(A) = 0.12, P(B) = 0.4. Find P(A|B)