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: 22 Given P(4) =0.5, P(B) = 0.6 and P(AU B) = 0.8. Calculate the probability that both events A and B…

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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 two events on the same sample space. The probability that the event A occurs equals…

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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: Q. 4 Let A, B, and C, be independent events with probabilities pa, Pb. Pc of occurring,…

A: The probabilities of 3 events A, B and C are defined as Pa, Pb and Pc respectively. Since the…

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: 1. Let A and B be events with probabilities P(A) = and P(B) = . Show that ≤P(ANB) ≤ }}, and give…

A: It is given that P(A)=34 and P(B)=13.

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…

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Q: at random events A and Bare independent, A and C are independent, and BC = 6. %3D P(B) =,P(AC |…

A: Given that

Q: Two events A and B are such that P(A) = 0.19, P(AN B)= 0.15, P(A or B ) = 0.99. Find P(A N BC).…

A: The given events and their probabilities are as follows: P(A)=0.1 PA∩B=0.15 PA∪B=0.99 Need to find…

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: Q2:Two dice are thrown together. Let A be the event getting less than 3 (A2) on the second die. Are…

A: Given that Two dice are thrown together. Sample space=S={(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),…

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…

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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: Q. 4 Let A, B, and C, be independent events with probabilities Pa, Pb, Pc of occurring,…

A: It is an important part of statistics. It is widely used.

Q: Assume that the events A and B are independent with P(A) = .30 and P(B) = 40. The probability that…

A: We have given that,  P(A) = 0.30 and P(B) = 0.40 Then,  We will find the probability that  both…

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.…

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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…

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

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Q: you say about P(B)?

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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: Two events A and B are such that P(A /B) = 0.5, P(A N B)= 0.15,P(A or B ) = 0.65. Find P(ACN B©).…

A: From the provided information, P (A|B) = 0.5, P (A Ո B) = 0.15, P (A or B) = 0.65

Q: Given two events A and B such that P (A) = 1/2 and P (B) = 1/3. Determine P (B | A) if   (a) A and B…

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Q: Q. 4 Let A, B, and C, be independent events with probabilities pa, Pp. Pc of occurring,…

A: From the given information, P(A)=pa P(B)=pb P(C)=pc i) The probability all 3 events occurring is,…

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)

Q: Show that the probability that an event A or an event B or both will occur is given by the formula…

A: Probability of an event A = P(A) Probability of occurring event B = P(B) Probability of occurring A…

Q: Suppose P and Q are two independent events with P(P) = 0.23, P(P and Q)= 0.18 Calculate P

A: P(P) = 0.23, P(P and Q)= 0.18 P and Q are independent events