Understanding Mutually Exclusive and Independent Events in Probability

This question asks about the relationship between two events, A and B, that are described as mutually exclusive and have positive probabilities P(A) > 0 and P(B) > 0. We need to determine if they are independent or not.

Defining Key Probability Concepts

Let's first clarify the definitions of the terms used:

• Mutually Exclusive Events: Two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time. In set theory terms, their intersection is the empty set, <code>\(A \cap B = \emptyset\)</code>. The probability of their intersection is always zero: <code>\(P(A \cap B) = 0\)</code>.
• Independent Events: Two events A and B are independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, events A and B are independent if and only if the probability of their intersection is equal to the product of their individual probabilities: <code>\(P(A \cap B) = P(A) \times P(B)\)<br /></code>An alternative definition is <code>\(P(A|B) = P(A)\)</code> (if <code>\(P(B) > 0\)) or <code>\(P(B|A) = P(B)\)</code> (if <code>\(P(A) > 0\)).

Analyzing the Relationship between Mutually Exclusive and Independent Events

We are given that events A and B are mutually exclusive and that \( P(A) > 0 \) and \( P(B) > 0 \).

Since A and B are mutually exclusive, we know that:

<code>\(P(A \cap B) = 0\)</code>

Now let's consider the condition for independence:

<code>\(P(A \cap B) = P(A) \times P(B)\)</code>

For A and B to be independent, their intersection probability must be equal to the product of their individual probabilities. Let's substitute the value of \( P(A \cap B) \) for mutually exclusive events into the independence equation:

<code>\(0 = P(A) \times P(B)\)</code>

However, the problem states that \( P(A) > 0 \) and \( P(B) > 0 \). This means that the product \( P(A) \times P(B) \) must be a positive value:

<code>\(P(A) \times P(B) > 0\)</code>

Comparing the two results, we see that:

<code>\(P(A \cap B) = 0\)</code> (because A and B are mutually exclusive)

<code>\(P(A) \times P(B) > 0\)</code> (because \( P(A) > 0 \) and \( P(B) > 0 \))

Since \( 0 \neq P(A) \times P(B) \), the condition for independence \( P(A \cap B) = P(A) \times P(B) \) is not satisfied.

Therefore, if A and B are mutually exclusive events with positive probabilities, they cannot be independent.

In fact, if two events A and B with \( P(A) > 0 \) and \( P(B) > 0 \) are mutually exclusive, the occurrence of event A makes the occurrence of event B impossible (since they cannot happen at the same time), and vice versa. This strong dependence means they are not independent.

Evaluating the Given Options

Let's look at the options based on our analysis:

1. A and B are independent: Incorrect, as shown above, they are not independent if \( P(A) > 0 \) and \( P(B) > 0 \).
2. A and B are not independent: Correct, this aligns with our conclusion.
3. \( A \subseteq B \): This means A is a subset of B. If A is a non-empty subset of B, then the intersection \( A \cap B \) would be A, and \( P(A \cap B) = P(A) \). If A and B are also mutually exclusive, their intersection must be empty (\( P(A \cap B) = 0 \)). For both to be true, \( P(A) \) would have to be 0. But we are given \( P(A) > 0 \). So, \( A \subseteq B \) cannot be true under the given conditions (unless A is the empty set, which contradicts \( P(A) > 0 \)).
4. \( B \subseteq A \): Similar to option 3, this would require \( P(B) = 0 \) for A and B to be mutually exclusive, which contradicts the condition \( P(B) > 0 \).

Based on the properties of mutually exclusive events with non-zero probabilities, they must be not independent.

Conclusion on Mutually Exclusive Events and Independence

The core difference is in the intersection. Mutually exclusive events have an intersection of probability 0. Independent events have an intersection probability that is the product of their individual probabilities. If both P(A) and P(B) are positive, the product \( P(A) \times P(B) \) is also positive. Since 0 cannot equal a positive number, two events that are mutually exclusive with positive probabilities cannot satisfy the condition for independence. They are necessarily dependent.

Revision Table: Key Probability Event Types

Additional Information: When Mutually Exclusive Events CAN be Independent

The only exception to the rule that mutually exclusive events with non-zero probabilities are not independent is when at least one of the events has a probability of zero. If, for example, \( P(A) = 0 \) or \( P(B) = 0 \), then:

• \( P(A \cap B) = 0 \) (This is true if they are mutually exclusive, or if A or B is an impossible event).
• \( P(A) \times P(B) = 0 \times P(B) = 0 \) (if \( P(A) = 0 \)), or \( P(A) \times P(B) = P(A) \times 0 = 0 \) (if \( P(B) = 0 \)).

In this special case where \( P(A \cap B) = 0 \) and \( P(A) \times P(B) = 0 \), the condition for independence \( P(A \cap B) = P(A) \times P(B) \) is satisfied (0 = 0). So, if either \( P(A)=0 \) or \( P(B)=0 \), mutually exclusive events are also independent. However, the question specifically states \( P(A) > 0 \) and \( P(B) > 0 \), which excludes this special case.