Certainly! Let's go through the concepts step by step to identify which equation represents the condition that the events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] are nonexclusive.
### Definitions:
1. Nonexclusive (Non-mutually exclusive) Events: Events that can occur together. This means they have a non-empty intersection.
2. Exclusive (Mutually exclusive) Events: Events that cannot occur together. This means their intersection is empty.
### Analyzing the Conditions:
1. Option A: [tex]\( P(E_1 \cap E_2) \neq 0 \)[/tex]
- This condition says that the probability of both events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] happening together is not zero.
- If [tex]\( P(E_1 \cap E_2) \neq 0 \)[/tex], it means that there is a positive probability that both events can occur simultaneously.
- Nonexclusive Events can occur together, so their intersection probability should not be zero.
2. Option B: [tex]\( P(E_1 \cup E_2) = 1 \)[/tex]
- This condition says that the probability of either event [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] or both happening is equal to 1.
- This condition implies that at least one of the events must happen.
- This doesn't specifically address the nonexclusivity of the events, but rather a certainty of one event occurring.
3. Option C: [tex]\( P(E_1 \cap E_2) = 0 \)[/tex]
- This condition says that the probability of both events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] happening together is zero.
- If [tex]\( P(E_1 \cap E_2) = 0 \)[/tex], it means the events are mutually exclusive; they cannot occur together.
4. Option D: [tex]\( P(E_1 \cup E_2) \neq 1 \)[/tex]
- This condition says that the probability of either event [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] or both happening is not equal to 1.
- This condition indicates that it's not certain that one of the events will occur but doesn't specifically say anything about their nonexclusivity.
### Conclusion:
- From the definitions and conditions, the correct representation for nonexclusive events (i.e., events that can occur together) is given by the condition:
[tex]\[
P(E_1 \cap E_2) \neq 0
\][/tex]
Thus, the correct answer is Option A.
