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To understand what the probability [tex]\( P\left(E_1 \cup E_2\right) \)[/tex] represents, it is important first to understand what the notation [tex]\( \cup \)[/tex] signifies in probability theory.
The symbol [tex]\( \cup \)[/tex] is called the union of two events. For any two events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex], the union [tex]\( E_1 \cup E_2 \)[/tex] consists of all outcomes that are in [tex]\( E_1 \)[/tex], in [tex]\( E_2 \)[/tex], or in both [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex]. Therefore, the union represents the event that at least one of the events [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] happens. It includes the scenarios where:
- Only [tex]\( E_1 \)[/tex] occurs,
- Only [tex]\( E_2 \)[/tex] occurs,
- Both [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occur simultaneously.
Let's examine the given options one by one:
A. One of the events occurs but not both.
- This describes the situation where either [tex]\( E_1 \)[/tex] occurs and [tex]\( E_2 \)[/tex] does not, or [tex]\( E_2 \)[/tex] occurs and [tex]\( E_1 \)[/tex] does not. This does not account for the scenario where both events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occur together.
B. Both of the events occur.
- This describes the intersection of events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] which is denoted by [tex]\( E_1 \cap E_2 \)[/tex]. This does not include the scenarios where only [tex]\( E_1 \)[/tex] occurs or only [tex]\( E_2 \)[/tex] occurs.
C. One of the events occurs, or both occur.
- This accurately describes the union [tex]\( E_1 \cup E_2 \)[/tex]. It includes all scenarios where at least one of the events happens, and it incorporates the cases where both events happen too.
D. Neither event occurs.
- This describes the complement of the union [tex]\( (E_1 \cup E_2)^c \)[/tex]. This means neither [tex]\( E_1 \)[/tex] nor [tex]\( E_2 \)[/tex] happens, which is not what [tex]\( P(E_1 \cup E_2) \)[/tex] represents.
Given this understanding, the correct interpretation of the probability [tex]\( P\left(E_1 \cup E_2\right) \)[/tex] is:
C. One of the events occurs, or both occur.
The symbol [tex]\( \cup \)[/tex] is called the union of two events. For any two events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex], the union [tex]\( E_1 \cup E_2 \)[/tex] consists of all outcomes that are in [tex]\( E_1 \)[/tex], in [tex]\( E_2 \)[/tex], or in both [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex]. Therefore, the union represents the event that at least one of the events [tex]\( E_1 \)[/tex] or [tex]\( E_2 \)[/tex] happens. It includes the scenarios where:
- Only [tex]\( E_1 \)[/tex] occurs,
- Only [tex]\( E_2 \)[/tex] occurs,
- Both [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occur simultaneously.
Let's examine the given options one by one:
A. One of the events occurs but not both.
- This describes the situation where either [tex]\( E_1 \)[/tex] occurs and [tex]\( E_2 \)[/tex] does not, or [tex]\( E_2 \)[/tex] occurs and [tex]\( E_1 \)[/tex] does not. This does not account for the scenario where both events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occur together.
B. Both of the events occur.
- This describes the intersection of events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] which is denoted by [tex]\( E_1 \cap E_2 \)[/tex]. This does not include the scenarios where only [tex]\( E_1 \)[/tex] occurs or only [tex]\( E_2 \)[/tex] occurs.
C. One of the events occurs, or both occur.
- This accurately describes the union [tex]\( E_1 \cup E_2 \)[/tex]. It includes all scenarios where at least one of the events happens, and it incorporates the cases where both events happen too.
D. Neither event occurs.
- This describes the complement of the union [tex]\( (E_1 \cup E_2)^c \)[/tex]. This means neither [tex]\( E_1 \)[/tex] nor [tex]\( E_2 \)[/tex] happens, which is not what [tex]\( P(E_1 \cup E_2) \)[/tex] represents.
Given this understanding, the correct interpretation of the probability [tex]\( P\left(E_1 \cup E_2\right) \)[/tex] is:
C. One of the events occurs, or both occur.
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