Question

select the correct answer.
two events, ( e_1 ) and ( e_2 ), are defined for a random experiment. which equation represents the condition that the events are nonexclusive?
a. ( p(e_1 cap e_2)=0 )
b. ( p(e_1 cup e_2)
eq1 )
c. ( p(e_1 cup e_2)=1 )
d. ( p(e_1 cap e_2)
eq0 )

Explanation:

Step1: Recall the definition of non - exclusive events

Non - exclusive events are events that can occur at the same time. In probability terms, if two events \(E_1\) and \(E_2\) are non - exclusive, then there is an overlap between them.
The probability of the intersection of two events \(P(E_1\cap E_2)\) represents the probability that both events occur simultaneously.

Step2: Analyze each option

• Option A: \(P(E_1\cap E_2) = 0\) means the events are mutually exclusive (no overlap).
• Option B: \(P(E_1\cup E_2)

eq1\) just tells us about the probability of the union not being equal to 1, which is not the key property for non - exclusivity.

• Option C: \(P(E_1\cup E_2)=1\) means the union of the two events is the entire sample space, which is not related to the non - exclusivity of the two events themselves.
• Option D: \(P(E_1\cap E_2)

eq0\) implies that there is a non - zero probability of both events \(E_1\) and \(E_2\) occurring at the same time, which is the definition of non - exclusive events.

Answer:

D. \(P(E_1\cap E_2)
eq0\)