Question

If one event is unaffected by the outcome of another event, the two events are said to be:

• A. Mutually exclusive
• B. Dependent
• C. Either dependent or independent
• D. Independent

Hint:

Independent events are defined by the fact that their probabilities remain unaffected by each other. Remember, \( P(A \cap B) = P(A) \cdot P(B) \) for independence.

Answer 1

The Correct Option is D

Step 1: Understanding independent events. Two events are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other. Mathematically, for two independent events \( A \) and \( B \): \[ P(A \cap B) = P(A) \cdot P(B) \] 

Step 2: Explanation of incorrect options. 
- (A) Mutually exclusive: Mutually exclusive events cannot occur simultaneously. If one event occurs, the other cannot, making them dependent in a way. 
- (B) Dependent: Dependent events are those where the occurrence of one event affects the occurrence of the other. 
- (C) Either dependent or independent: This is ambiguous and does not define the relationship clearly.

Answer 2

The question describes two events where the outcome of one has no effect on the outcome of the other, and asks what this relationship is called. Let's check each term against that description.

1. Mutually exclusive: Mutually exclusive events cannot both happen at the same time, meaning if one occurs, the other is automatically ruled out. This is a relationship between outcomes, not the "no effect on each other" description given in the question.
2. Dependent: Dependent events are exactly the opposite of what is described, since the outcome of one event changes the probability of the other. That contradicts "unaffected by the outcome."
3. Either dependent or independent: This answer does not commit to a single relationship and fails to name the specific term the question is asking for.
4. Independent: Two events are called independent precisely when knowing the outcome of one gives no information about the other, matching the wording in the question exactly. For independent events \( A \) and \( B \), \[ P(A \cap B) = P(A) \cdot P(B) \] because neither event's probability shifts based on the other happening.

The definition in the question, one event being unaffected by another, is the textbook definition of independence between events.

So the correct answer is Independent.