Let's break down the notation and concepts step-by-step to understand what the probability [tex]\( P(E_1 \cup E_2) \)[/tex] represents.
1. Probability Basics:
- Probability, in general, measures the likelihood of an event occurring.
- Probabilities range from 0 to 1, where 0 means the event definitely does not occur, and 1 means the event definitely occurs.
2. Events and Union of Events:
- An event [tex]\( E \)[/tex] is something that can occur in a probability space (e.g., getting a heads in a coin toss).
- The union of two events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex], denoted [tex]\( E_1 \cup E_2 \)[/tex], represents the event that either [tex]\( E_1 \)[/tex] occurs, [tex]\( E_2 \)[/tex] occurs, or both occur.
3. Interpreting [tex]\( P(E_1 \cup E_2) \)[/tex]:
- [tex]\( P(E_1 \cup E_2) \)[/tex] specifically measures the probability that at least one of the events [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] occurs.
- "At least one" includes three possible scenarios:
- Only [tex]\( E_1 \)[/tex] occurs.
- Only [tex]\( E_2 \)[/tex] occurs.
- Both [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occur.
From this explanation, it's clear that:
- Option A (Both of the events occur) is incorrect because it implies that only the scenario where both events occur is considered.
- Option B (One of the events occurs but not both) is incorrect because it excludes the scenario where both events occur.
- Option C (Neither event occurs) is incorrect because it represents the opposite of what [tex]\( E_1 \cup E_2 \)[/tex] stands for.
- Option D (One of the events occurs, or both occur) correctly represents the union of the two events.
Therefore, the answer is D. One of the events occurs, or both occur.
