Write coherent proofs for the following basic consequences of the axioms of probability theory:
a. P(Ac)=1−P(A)usingS=A∪Ac disjoint union; b. 0≤P(A)≤1; c. P(∅)=0
Write coherent proofs for the following basic consequences of the axioms of probability theory: a. P(Ac)=1−P(A)usingS=A∪Ac...
Proofs a) With conditional probability, P(A|B), the axioms of probability hold for the event on the left side of the bar. A useful consequence is applying the complement rule to conditional probability. We have that P(A|B) = 1 − P(A|B). Prove this by showing that P(A|B) + P(A|B) = 1 (Hint: just use the definition of conditional probability) b) If two events A and B are independent, then we know P(A ∩ B) = P(A)P(B). A fact is that if...
One of the basic requirements of probability is ______ Note: P(Ac) in the miltiple choice answer = P(Ac) A. for each experimental outcome Ei, we must have P(Ei) ≥ 1 B. if there are k experimental outcomes, then P(E1) + P(E2) + ... + P(Ek) = 1 C both P(A) = P(Ac) − 1 and if there are k experimental outcomes, then P(E1) + P(E2) + ... + P(Ek) = 1 D P(A) = P(Ac) − 1
2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) If AnC 0 (empty set), then P[An CIB] P[AIB] + P[CIB] 2. Show that P[AIB] satisfies the three axioms of probability b) PISIB] 1 for sample space S c) If AnC 0 (empty set), then P[An CIB] P[AIB] + P[CIB]
2. Prove the following propositions (a) Proposition 1: For every event A, AC A (b) Proposition 2: If A, B, C are events, if A c B and if Bc C, then Ac C (c) Proposition 3: φ-Ω and 0° = φ (d) Proposition 4: If A1, ..Ak (e) Proposition 5: If A and Bare events, then P(A UB)-P(A)+P(B) - P(AB) are disjoint events, then P(UK 1 A.)-Σ'm P(A)
2. Suppose that some agent whose degrees of belief are coherent ascribes subiective probabilities to propositions P and Q as follows: Use the laws of probability to compute each of the following b) Pr(PvQ) e Pr (PQ) d) Pr(Q P) f) Pr(- P Q) IMPORTANT: You may assume that none of x, y, and z has a value of either 0 or 1, but do not assume that P and Q are probabilistically independent. 2. Suppose that some agent whose...
any help with these problems? 0 2 pts ect Question 13 The addition rule for probability P(A U B) for: p(A) + P(B)-PA n В) is used finding the probability that A happens, then B happens. hinding the probability that A doesn't happen, but B does happen. finding the probability that A or B or both happen 9 inding the probability that A and B both happen Quiz Score: 5.8 out of Question 12 0/ 2 pts The multiplication rule...
1. Find P(AU(B UC)) in each of the following four cases: (a) A, B, and C are disjoint events and P(A) 1/2. (b) P(A)-2P(BC)= 3P(ABC) =1/2 (c) P(A)1/2, P(BC) 1/3, and P(AC)0 d) P(An (Be UC))-0.7
Question 1: Basic Probability Theory A manufacturing facility is responsible for producing two components that are produced at states 1 (poor), 2 (below average), 3 (average), 4 (good), and 5 (excellent). For the final product (con sisting of both components) to be acceptable the sum of the two states needs to be at least equal to 6. At the mmen, the manufacturing processes are random, and hence each component is equally likely to be at any of the 5 states....
probability help (1 point) Assume that P(A n Bº) = 0.29, P(B n AC) = 0.2, and P((A U B)") = 0.12. Find P(An B). P(A n B) =
Find P(A U (Be UC)9) in each of the following four cases: (a) A, B, and C are disjoint events and P(A) 1/2. (b) P(A)2P(BC)3P(ABC)-1/2 (c) P(A)1/2, P(BC) 1/3, and P(AC)0 (d) PA n (BC UC) 0.7