To determine whether two events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] are nonexclusive, we need to understand what "nonexclusive" means in probability theory.
Two events are considered nonexclusive if they can both occur simultaneously. In other words, their intersection is non-empty. This means that there is a non-zero probability that both [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] happen at the same time.
Let's analyze the given options:
A. [tex]\( P\left(E_1 \cap E_2\right)=0 \)[/tex]
If this equation holds, it means that the probability of both events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occurring together is zero. This implies that the events are mutually exclusive, not nonexclusive. Therefore, this option is incorrect.
B. [tex]\( P\left(E_1 \cup E_2\right) \neq 1 \)[/tex]
This option suggests that the probability that either [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] (or both) occur is not equal to one. This condition does not specifically address whether the events can occur simultaneously. Hence, this option is also incorrect.
C. [tex]\( P\left(E_1 \cap E_2\right) \neq 0 \)[/tex]
If this equation holds, it means that the probability of both events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occurring together is not zero. This indicates that there is a non-zero chance for both events to happen simultaneously, implying that the events are nonexclusive. This option correctly represents the condition we're looking for.
D. [tex]\( P\left(E_1 \cup E_2\right)=1 \)[/tex]
This equation means that the probability that either [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] (or both) occur is equal to one. While this can be true for nonexclusive events, it does not explicitly indicate the nature of the intersection of [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex]. Therefore, this option does not uniquely define nonexclusive events and is thus incorrect.
Therefore, the correct answer is:
[tex]\[ \boxed{C} \quad P\left(E_1 \cap E_2\right) \neq 0 \][/tex]
