Sure, let's analyze the addition theorem and its applicability step by step.

The addition theorem in probability is generally stated as follows:

For any two events A and B, the probability of A or B occurring (denoted as P(A ∪ B)) is given by:
$$ P(A ∪ B) = P(A) + P(B) - P(A ∩ B) $$

This theorem is used to find the probability of the union of two events. Now, let's consider each of the given options:

1. **Mutually exclusive events**:
- Mutually exclusive events are events that cannot occur at the same time. In other words, if one event occurs, the other cannot.
- For mutually exclusive events, $ P(A ∩ B) = 0 $.
- The addition theorem simplifies to $ P(A ∪ B) = P(A) + P(B) $.
- Therefore, the addition theorem is applicable to mutually exclusive events.

2. **Exhaustive events**:
- Exhaustive events are a set of events that cover all possible outcomes of an experiment.
- The addition theorem can still be applied to exhaustive events, but it doesn't specifically require the events to be exhaustive.
- Therefore, while the addition theorem can be used with exhaustive events, it is not specifically designed for them.

3. **Conditional events**:
- Conditional events are events where the probability of one event depends on the occurrence of another event.
- The addition theorem does not specifically address conditional probabilities. Instead, it deals with the probability of the union of two events.
- Therefore, the addition theorem is not specifically applicable to conditional events.

4. **Independent events**:
- Independent events are events where the occurrence of one event does not affect the probability of the other event.
- For independent events, $ P(A ∩ B) = P(A) \times P(B) $.
- The addition theorem can still be applied to independent events, but it is not specifically designed for them.
- Therefore, while the addition theorem can be used with independent events, it is not specifically applicable to them.

Based on the analysis, the addition theorem is specifically applicable to **mutually exclusive events**.

Question

Sure, let's analyze the addition theorem and its applicability step by step.

The addition theorem in probability is generally stated as follows:

For any two events A and B, the probability of A or B occurring (denoted as P(A ∪ B)) is given by:
$$ P(A ∪ B) = P(A) + P(B) - P(A ∩ B) $$

This theorem is used to find the probability of the union of two events. Now, let's consider each of the given options:

1. **Mutually exclusive events**:
   - Mutually exclusive events are events that cannot occur at the same time. In other words, if one event occurs, the other cannot.
   - For mutually exclusive events, $ P(A ∩ B) = 0 $.
   - The addition theorem simplifies to $ P(A ∪ B) = P(A) + P(B) $.
   - Therefore, the addition theorem is applicable to mutually exclusive events.

2. **Exhaustive events**:
   - Exhaustive events are a set of events that cover all possible outcomes of an experiment.
   - The addition theorem can still be applied to exhaustive events, but it doesn't specifically require the events to be exhaustive.
   - Therefore, while the addition theorem can be used with exhaustive events, it is not specifically designed for them.

3. **Conditional events**:
   - Conditional events are events where the probability of one event depends on the occurrence of another event.
   - The addition theorem does not specifically address conditional probabilities. Instead, it deals with the probability of the union of two events.
   - Therefore, the addition theorem is not specifically applicable to conditional events.

4. **Independent events**:
   - Independent events are events where the occurrence of one event does not affect the probability of the other event.
   - For independent events, $ P(A ∩ B) = P(A) \times P(B) $.
   - The addition theorem can still be applied to independent events, but it is not specifically designed for them.
   - Therefore, while the addition theorem can be used with independent events, it is not specifically applicable to them.

Based on the analysis, the addition theorem is specifically applicable to **mutually exclusive events**.

Knowee AI · Accepted Answer