Question

select the correct answer.
given any two events, $e_1$ and $e_2$, what does the probability $p(e_1 \cup e_2)$ represent?

a. one of the events occurs but not both.
b. both of the events occur.
c. one of the events occurs, or both occur.
d. neither event occurs.

Explanation:

In probability theory, the union of two events \( E_1 \) and \( E_2 \), denoted as \( E_1 \cup E_2 \), represents the set of outcomes that are in \( E_1 \), in \( E_2 \), or in both. So \( P(E_1 \cup E_2) \) is the probability that one of the events occurs (either \( E_1 \) or \( E_2 \)) or both occur.

• Option A describes the probability of the symmetric difference (one or the other but not both), which is \( P((E_1 - E_2) \cup (E_2 - E_1)) \), not the union.
• Option B describes the intersection \( P(E_1 \cap E_2) \), the probability both occur.
• Option D describes the complement of the union, \( P((E_1 \cup E_2)^c) \), the probability neither occurs.

Answer:

C. One of the events occurs, or both occur.