In all of the following questions, let be a sample space with a probability measure P, and let A, BC be events. B1. (a) Starting from just the three probability axioms, prove that P(0) = 0. (b) Let A and B be two events with P(A) = 0.8 and P(B) = 0.4. Prove the upper bound P(ANB) ≤ 0.4. You may use any of the properties of probability stated in the lecture notes. (c) Prove that the upper bound in (b) can be achieved, by giving an example of a sample space, a probability measure P and events A, BC such that P(A) = 0.8, P(B) = 0.4 and P(ANB) = 0.4.

Q: Which of the following is NOT necessarily true for mutually exclusive events A and B? P(A B)=0…

A: If A and B are mutually exclusive events, then A∩B=∅. Then, we have P(A∩B)=P(AB)=0. 1.)…

Q: Let A and B be events where P(A)= .3, P(B)= .8, and P(A or B)= .9. Compute P( A and B).

A: The objective of this question is to find the probability of both events A and B occurring, denoted…

Q: For not mutually exclusive events A and B, P(A) = 0.59, P(B)=0.46 and P(A B)=0.28. Find the P(AUB).

A: Given:        The A and B are not mutually exclusive P(A) = 0.59 P(B) = 0.46 P(A ∩ B) = 0.28

Q: Let A, B and C be events in a sample space S such that S= AUB. Suppose that P(A) = 0.4, P(B) = 0.5,…

A: Given that P(A)=0.4,P(B)=0.5 and P(A∩B)=0.4 Then the aim is to find P(Ac∪Bc)

Q: Every customer in a restaurant is served water. I represents a randomly selected customer that…

A: Given events; I: a randomly selected customer that has water with ice. L: a randomly selected…

Q: If A  and   B    are independent events with    P(A)=0.24    and    P(B)=0.73  Calculate and find…

A:

Q: If A and B are events with P(A)=0.6, P(A OR B)=0.98, P(A AND B)=0.02, find P(B).

A:

Q: Suppose that C and D are events such that P(C) = 4/5 and P(C&D) = 3/10. Find P(D| C).

A: The formula for conditional probability is,

Q: If P(A)=0.3 and P(B)=0.6, find each probability  a) P(A and B) if A and B are independent events…

A:

Q: For two events, A and B, we have P(A) = 0.8, P(B) = 0.7, and P(AUB) = 0.9. Then, the value for P(BA)…

A: Given that  P(A) = 0.8, P(B) = 0.7, and P(A U B) = 0.9.

Q: P(lightning | rain) = P(rain | lightning):

A: We have given that  The total number of days = 365 Rainy days = 103 Lightning days = 20 Rainy and…

Q: Given P(A) = 0.4 and P(B) = 0.6, do the following. a) f A and B are independent events, compute P(A…

A: a)Recall that if two events are independent, then the probability of their intersection, P(AandB),…

Q: Suppose that A,B, & C are three independent events such that Pr(A)=1/4, Pr(B)=1/3, and Pr(C)=1/2. A.…

A:

Q: 3) Let us assume that we have a five sided die. The sides of the die are marked with the numbers –2,…

A: The distribution of X is given  Let us compute E(X2)  E(X2)=∑xx2P(X=x)=(-2)2 P(X=-2)…

Q: Let A and B be events such that P (A) = 1/5 P (A & B) = 1/10 and P (A or B) = 1/2 Determine P (B)

A: P(A)=15P(A and B) = 110P(A or B) = 12

Q: Suppose that A and B are two independent events for which P(A)=0.32 and P(B)=0.7. Find each of the…

A:

Q: Assume that events X, Y, Z are mutually exclusive (pairwise) and P(X) = 0.132 P(Y) = 0.1618 P(Z)…

A:

Q: you say about P(B)?

A:

Q: Let A and B be any two events defined on S. suppose that p(A)=0.2, p(B)=0.7, and p(A and B)=0.3, the…

A: It is given that, p(A) =0.2 p(B) =0.7 and p(A and B) =0.3

Q: 5.Prove the following properties of probability: P(A) = 1-P(A). b. For any events A and B, P(A U B)…

A: Solution-:We want to prove the following properties of probability:(a) (b)