Let A and B be events. Find an expression and exhibit the Venn diagram for the event that: (a) A or not B occurs, (b) only A occurs.

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**Event Description and Venn Diagrams**

In probability theory, we often deal with events, which are sets of outcomes from some experiment. Let \( A \) and \( B \) be events. We are tasked with finding expressions and visualizing these events using Venn diagrams for the following scenarios:

1. **\( A \) or not \( B \) occurs:**
   - **Expression:** This can be expressed as \( A \cup \overline{B} \).
   - **Explanation:** The event \( A \cup \overline{B} \) consists of all outcomes that are either in \( A \) or not in \( B \). In a Venn diagram, this includes the entire region of circle \( A \) and the area outside circle \( B \).

2. **Only \( A \) occurs:**
   - **Expression:** This is expressed as \( A \cap \overline{B} \).
   - **Explanation:** The event \( A \cap \overline{B} \) includes all outcomes that are in \( A \) but not in \( B \). In a Venn diagram, it represents the portion of circle \( A \) that does not overlap with circle \( B \).

These Venn diagrams help to visualize the relationships between different sets (events) and to understand how the operations \( \cup \) (union) and \( \cap \) (intersection) work with complements (\(\overline{B}\)).
Transcribed Image Text:**Event Description and Venn Diagrams** In probability theory, we often deal with events, which are sets of outcomes from some experiment. Let \( A \) and \( B \) be events. We are tasked with finding expressions and visualizing these events using Venn diagrams for the following scenarios: 1. **\( A \) or not \( B \) occurs:** - **Expression:** This can be expressed as \( A \cup \overline{B} \). - **Explanation:** The event \( A \cup \overline{B} \) consists of all outcomes that are either in \( A \) or not in \( B \). In a Venn diagram, this includes the entire region of circle \( A \) and the area outside circle \( B \). 2. **Only \( A \) occurs:** - **Expression:** This is expressed as \( A \cap \overline{B} \). - **Explanation:** The event \( A \cap \overline{B} \) includes all outcomes that are in \( A \) but not in \( B \). In a Venn diagram, it represents the portion of circle \( A \) that does not overlap with circle \( B \). These Venn diagrams help to visualize the relationships between different sets (events) and to understand how the operations \( \cup \) (union) and \( \cap \) (intersection) work with complements (\(\overline{B}\)).
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