Can two events with nonzero probabilities be both independent and mutually exclusive? Choose the correct answer below. A. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities add up to one. B. No, two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually exclusive, then when one of them occurs, the probability of the other must be zero. C. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal. D. No, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.
The Correct Answer and Explanation is:
Correct Answer:
B. No, two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually exclusive, then when one of them occurs, the probability of the other must be zero.
Explanation:
In probability theory, the concepts of independence and mutual exclusivity are fundamentally different and, in fact, incompatible for events with nonzero probabilities.
Two events are said to be mutually exclusive if they cannot occur at the same time. That is, if event A happens, event B cannot happen, and vice versa. Mathematically, this means that the probability of both A and B occurring together is zero. This is written as:P(A∩B)=0P(A \cap B) = 0P(A∩B)=0
On the other hand, two events are said to be independent if the occurrence of one does not affect the probability of the other. In other words, knowing that one event occurred does not provide any information about whether the other occurred. The formal definition of independence is:P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B)P(A∩B)=P(A)⋅P(B)
Now, if two events A and B are mutually exclusive and each has a nonzero probability, then their intersection is zero, as stated above. However, for independence to hold, the probability of their intersection must equal the product of their individual probabilities. If both events have nonzero probabilities, then their product is also nonzero. This leads to a contradiction because the left-hand side is zero (due to mutual exclusivity) while the right-hand side is positive (due to independence). Therefore, both conditions cannot be true at the same time when the events have nonzero probabilities.
In conclusion, two events with nonzero probabilities cannot be both independent and mutually exclusive. The requirement that their joint probability be both zero and nonzero is logically impossible.
