Can two events with nonzero probabilities be both independent and mutually exclusive? Explain your reasoning.

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Can two events with nonzero probabilities be both independent and mutually exclusive? Explain your  reasoning.

Szeteib · Accepted Answer

If two events cannot occur at the same moment, they are mutually exclusive. Two occurrences are considered independent if the occurrence of one of them has no effect on the occurrence of the other. Thus, two events cannot occur simultaneously because if one of the occurrences occurs, we know that the other event does not occur, and thus the second event is influenced by the first event occurring.Result: No

RizerMix · Accepted Answer

No, two events with nonzero probabilities cannot be both independent and mutually exclusive.To see why, let A and B be two events with nonzero probabilities such that they are independent and mutually exclusive. Then, by definition:P(A∩B)=P(A)·P(B)  (independence)andP(A∩B)=0  (mutual exclusivity)However, these two statements are contradictory since P(A∩B)=0 means that the events A and B have no common outcomes, whereas P(A)·P(B)&amp;gt;0 means that the events are related in some way.Therefore, it is impossible for two events with nonzero probabilities to be both independent and mutually exclusive.

Jeffrey Jordon · Accepted Answer

Answer: NoExplanation:In probability theory, two events A and B are said to be independent if the occurrence of A does not affect the probability of B, and vice versa. In other words, P(A∩B)=P(A)P(B).On the other hand, two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time, i.e., P(A∩B)=0.Now, let&#039;s consider whether two events with nonzero probabilities can be both independent and mutually exclusive.Suppose that A and B are two events with nonzero probabilities such that they are both independent and mutually exclusive. Then, we have P(A∩B)=0 and P(A)P(B)≠0.Since P(A∩B)=0, we can write P(A∪B)=P(A)+P(B) by the addition rule for probability. Moreover, since A and B are independent, we have P(A∩B)=P(A)P(B). Combining these two equations, we get:P(A∪B)=P(A)+P(B)=P(A)+P(B)+2P(A)P(B)2P(A)P(B)=0This implies that either P(A)=0 or P(B)=0, which contradicts the assumption that both events have nonzero probabilities. Therefore, two events with nonzero probabilities cannot be both independent and mutually exclusive.

Vasquez · Accepted Answer

We know that two events A and B are said to be independent if the occurrence of A does not affect the occurrence of B, and vice versa. Mathematically, P(A∩B)=P(A)P(B).On the other hand, two events are mutually exclusive if they cannot occur simultaneously, i.e., P(A∩B)=0.Now, let&#039;s assume that two events A and B have nonzero probabilities, i.e., P(A)&amp;gt;0 and P(B)&amp;gt;0.If A and B are independent, then P(A∩B)=P(A)P(B)&amp;gt;0, which means that the intersection of A and B has a nonzero probability.But if A and B are mutually exclusive, then P(A∩B)=0, which contradicts our previous result.Therefore, it is impossible for two events with nonzero probabilities to be both independent and mutually exclusive.

Can two events with nonzero probabilities be both independent and mutually exclusive? Explain your reasoning.

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