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Suppose there are two events_ A and B_ Event A always happens (Pr[AJ=1) and event B is impossible (Pr[B]-0). Are events A and B independent of one another? Answers...

Question

Suppose there are two events_ A and B_ Event A always happens (Pr[AJ=1) and event B is impossible (Pr[B]-0). Are events A and B independent of one another? Answers are Yes, No, and Not enough information. You should also explain your answer This is a good prep for exams_

Suppose there are two events_ A and B_ Event A always happens (Pr[AJ=1) and event B is impossible (Pr[B]-0). Are events A and B independent of one another? Answers are Yes, No, and Not enough information. You should also explain your answer This is a good prep for exams_



Answers

Let A and B be two mutually exclusive events. Are A and B independent events? Explain your answer.

And this problem, we will show that two mutually exclusive events are not independent. Now, in the first part of the question, it is asked whether event can occur if event B has occurred. If A and B are mutually exclusive events now the N. B are mutually exclusive, it means that A and B cannot occur together, so if the event has occurred, then the answer is no. The event cannot occur because all of these events cannot occur together. Now the value of P. F. E. Given me, is what we need to determine next. The value of P. Of given B means the probability of the event, given that they even be has occurred since the even A cannot occur if they even be has occurred. That means that the probability of given B will be equal to zero because it is impossible for the event E to occur given that the event has occurred. Now, the next part of the question, we need to use this value of P. Of A. Even be to conclude that A. And B are not independent. Now, if we assume that A. And B are independent, then the value of P. L. E. And P is given to BP of a times PSP according to them an application. Now, it is also given that B. Of A is not equal to zero, and P F B is not equals to zero. So the product of two non zero numbers will be non zero. So that means that P. Of E and B is not equals to zero. However, the value of P of a given B is equal to zero. We have another multiplication rule which says that P O. E. And B is equals to the probability of E. Given B times value of P O. V. Now, we just determined that we have given B is equal to zero. So zero times P of P zero times B O B must be zero. So that means we have obtained the P R. E even be is not equals to zero, and again, we have obtained the P R E and B is equals to zero. Now this is a contradiction because both of these conditions cannot fall together. So that means that our assumption that P of E n B is equal to P times Tv. Because of our assumption that TNB are independent, is wrong. This part. This part must be wrong. So our assumption that A and B are independent events has to be wrong. So that means that A and B cannot be independent events if they are mutually exclusive.

So if we have events A. And B. That are destroying this means that they do not overlap, and now yes A. Or B. And C. Are also destroying this would be accurate. But then it says most events A. And C. Also beat this joint. Well, the answer is not necessarily because A. And B. Have to be this joined, and B. And C have to be this joint, but that doesn't mean that A. And C. Can be dis joined. So they had this A. And B. Would be destroyed, and and B. And C would be destroyed. But then and see would be this joint. So it's completely possible for A. And C. To be not this joint, while A. And C. Are both this joint with me.

Question a given and we are independent event. So probability is not equal to zero, and probability be is not all Sequent Tau zero. So the multiplication reward for the independent events at B A and the is ableto probably toe eight times probability off be so using that be a is not equal to zero and D B is not equal to zero. This means this is not equal to zero times you. So for ability of A and B is equal to preventive eight I be so the answer is yes.

Question eight Given A and B are mutually exclusive events s 02 mutually exclusive events that are two event that cannot occur at the same time. So if we know that event B has a cared, then we know that event. They cannot a care, since the events cannot occur at the same time. So an event to the public off a Kevin B is equal toe zero. Good to be, Yeah. Um, Aziz, we say that probability of a given B is equal to zero, so probability of a and B is equal toe. So the multiplication rule say that probability off a times probability off off the so as this is equal to zero. So it is not possible since we know that be a is not equal to zero. And the the is not equal to zero on this, then mean that the events are not independent. Since this, um, off the vacation rule for the independent event did not hold for this event. So we cannot apply this rule here in the two mutually exclusive weapons


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