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Select the correct answer.

Two events, [tex]E_1[/tex] and [tex]E_2[/tex], are defined for a random experiment. Which equation represents the condition that the events are nonexclusive?

A. [tex]P\left(E_1 \cap E_2\right) \neq 0[/tex]
B. [tex]P\left(E_1 \cup E_2\right)=1[/tex]
C. [tex]P\left(E_1 \cap E_2\right)=0[/tex]
D. [tex]P\left(E_1 \cup E_2\right) \neq 1[/tex]

Sagot :

GinnyAnswer
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.
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