1. **Problem Statement:** Identify 5 examples of non-mutually exclusive events and explain why they are non-mutually exclusive. 2. **Definition:** Non-mutually exclusive events are events that can occur at the same time, meaning their intersection is not empty. Formally, events $A$ and $B$ are non-mutually exclusive if $P(A \cap B) > 0$. 3. **Examples:** - Example 1: Drawing a card from a deck. Event $A$: drawing a red card. Event $B$: drawing a face card. These events are non-mutually exclusive because some cards (like the King of Hearts) are both red and face cards. - Example 2: Rolling a die. Event $A$: rolling an even number. Event $B$: rolling a number greater than 3. The numbers 4 and 6 satisfy both events, so they are non-mutually exclusive. - Example 3: Selecting a student. Event $A$: student is female. Event $B$: student is a senior. Some students can be both female and seniors. - Example 4: Weather events. Event $A$: it rains today. Event $B$: it is windy today. It can be both rainy and windy simultaneously. - Example 5: Tossing two coins. Event $A$: first coin is heads. Event $B$: second coin is heads. Both coins can be heads at the same time. 4. **Summary:** In all these examples, the events can happen together, so they are non-mutually exclusive. 5. **Formula for combined probability:** For two events $A$ and $B$, the probability of $A$ or $B$ occurring is $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$ This formula accounts for the overlap, which exists only if events are non-mutually exclusive. 6. **Explanation:** If events were mutually exclusive, then $P(A \cap B) = 0$, so the formula simplifies to $$ P(A \cup B) = P(A) + P(B) $$ because there is no overlap between the events.