Can two events with nonzero probabilities be both independent and mutually exclusive? Choose the correct answer below.
A. 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.
B. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities add up to one.
C. No, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.
D. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal.
The correct answer and explanation is :
The correct answer is A: “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:
To understand why A is the correct answer, we need to clarify the concepts of independence and mutual exclusivity.
- Mutually Exclusive Events: Two events (A) and (B) are mutually exclusive if they cannot happen at the same time. This means that if event (A) occurs, event (B) cannot occur, and vice versa. Mathematically, this is expressed as:
[
P(A \cap B) = 0
]
In other words, the probability of both events occurring together is zero. - Independent Events: Two events (A) and (B) are independent if the occurrence of one event does not affect the occurrence of the other event. This means that:
[
P(A \cap B) = P(A) \cdot P(B)
]
For the events to be independent, the probability of both events occurring together should be equal to the product of their individual probabilities.
Why A is correct:
If two events (A) and (B) are mutually exclusive, we know that:
[
P(A \cap B) = 0
]
However, if the events are independent, we must have:
[
P(A \cap B) = P(A) \cdot P(B)
]
For these two conditions to both hold true, it must be the case that:
[
P(A) \cdot P(B) = 0
]
This equation means that at least one of the probabilities (P(A)) or (P(B)) must be zero for the events to be both independent and mutually exclusive. Therefore, if both events have nonzero probabilities, they cannot be independent and mutually exclusive at the same time.
Why the other options are incorrect:
- B is incorrect because the sum of the probabilities of the two events being 1 does not change the fact that if events are mutually exclusive, (P(A \cap B) = 0), which conflicts with the requirement for independence that (P(A \cap B) = P(A) \cdot P(B)).
- C is incorrect because independence is not the complement of mutual exclusivity; they are distinct concepts that cannot both hold when the events have nonzero probabilities.
- D is incorrect because if the probabilities are equal, the events still cannot be both independent and mutually exclusive unless one of the events has zero probability.
The correct answer and explanation is :
The correct answer is A: “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:
To understand why A is the correct answer, we need to clarify the concepts of independence and mutual exclusivity.
- Mutually Exclusive Events: Two events (A) and (B) are mutually exclusive if they cannot happen at the same time. This means that if event (A) occurs, event (B) cannot occur, and vice versa. Mathematically, this is expressed as:
[
P(A \cap B) = 0
]
In other words, the probability of both events occurring together is zero. - Independent Events: Two events (A) and (B) are independent if the occurrence of one event does not affect the occurrence of the other event. This means that:
[
P(A \cap B) = P(A) \cdot P(B)
]
For the events to be independent, the probability of both events occurring together should be equal to the product of their individual probabilities.
Why A is correct:
If two events (A) and (B) are mutually exclusive, we know that:
[
P(A \cap B) = 0
]
However, if the events are independent, we must have:
[
P(A \cap B) = P(A) \cdot P(B)
]
For these two conditions to both hold true, it must be the case that:
[
P(A) \cdot P(B) = 0
]
This equation means that at least one of the probabilities (P(A)) or (P(B)) must be zero for the events to be both independent and mutually exclusive. Therefore, if both events have nonzero probabilities, they cannot be independent and mutually exclusive at the same time.
Why the other options are incorrect:
- B is incorrect because the sum of the probabilities of the two events being 1 does not change the fact that if events are mutually exclusive, (P(A \cap B) = 0), which conflicts with the requirement for independence that (P(A \cap B) = P(A) \cdot P(B)).
- C is incorrect because independence is not the complement of mutual exclusivity; they are distinct concepts that cannot both hold when the events have nonzero probabilities.
- D is incorrect because if the probabilities are equal, the events still cannot be both independent and mutually exclusive unless one of the events has zero probability.
