can two events with positive probabilities be both independent and mutually exclusive

Statistics
Can Two Events with Positive Probabilities be both Independent and Mutually Exclusive?
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Mutually exclusive events are ones that cannot both happen at the same time. In this case, the occurrence of an event A excludes the occurrence of event B. A perfect example is the tossing of a coin. At any given point, the coin can either be a head or a tail but not both.
P(A∩ B) = 0;P(A∪ B) = P (A) + P (B)P A B) = 0P (A │¬B)= P(A)/(1-P(B)On the other hand, independent events occur when the knowledge of the probability of occurrence of one does not change the possibility of the other occurring. This means that the occurrence of one of the events does not influence the possibility of occurrence of the other. In the same case of tossing a coin, the result of one of the flips does not affect the result of the other.
P(A∩ B)= P (A)P (B)P(A∪ B) = P (A) + P (B) – P (A)P (B)P (A ┤| B) = P (A)P (A │¬B)= P(A)It is possible to have both results of an analysis positive as the events followed in the analysis are independent of the first result. The probability of the second event giving similar result to the first is not affected by the initial probability meaning that the two events are independent (“Mutually Exclusive Events: Concepts, Solved & Practice Problems”, 2017).
This is the case of tossing two coins. In the case of a single coin, the occurrence of a head or a tail is mutually exclusive. Equally, the occurrence of a head prevents the occurrence of the tail meaning that the events in a single coin are dependent.

However, in the case of two coins or two tosses, the possibility of having a head in the first toss does not influence the possibility of having a head in the second event. It means that the two events can be mutually exclusive and independent.
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ReferencesMutually Exclusive Events: Concepts, Solved & Practice Problems. (2017). Retrieved from https://www.toppr.com/guides/quantitative-aptitude/probability/mutually-exclusive-events/