To determine the condition for two events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] to be nonexclusive, we first need to understand what nonexclusive events are.
Nonexclusive events are those that can occur simultaneously, meaning there is a possibility that both events can happen at the same time. In probability terms, this means that the intersection of the two events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] is not empty, i.e., there exists some outcome in the sample space that belongs to both events.
Mathematically, this condition can be represented as:
[tex]\[ P(E_1 \cap E_2) \neq 0 \][/tex]
Let's evaluate each given option to see which one matches this condition:
A. [tex]\( P(E_1 \cup E_2) = 1 \)[/tex]
- This condition states that the union of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] covers the entire sample space, meaning at least one of these events always happens. However, this does not specifically address whether [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] can occur together or not.
B. [tex]\( P(E_1 \cup E_2) \neq 1 \)[/tex]
- This condition states that the union of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] does not cover the entire sample space. Again, this condition does not provide information about the simultaneous occurrence of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex].
C. [tex]\( P(E_1 \cap E_2) \neq 0 \)[/tex]
- This condition states that the intersection of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] is not empty, meaning there is a nonzero probability that both [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] occur together. This is the condition we are looking for to define nonexclusive events.
D. [tex]\( P(E_1 \cap E_2) = 0 \)[/tex]
- This condition states that the intersection of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] is empty, meaning there is no possibility of both events occurring simultaneously. This describes exclusive (or mutually exclusive) events, which is the opposite of what we need.
Therefore, the correct answer is:
C. [tex]\( P(E_1 \cap E_2) \neq 0 \)[/tex]
