Explain mutually exclusive events and mutually exhaustive events with examples

### 1. Mutually Exclusive Events:

**Definition**: 
Two or more events are **mutually exclusive** if they cannot occur at the same time. This means the occurrence of one event prevents the other(s) from happening. These events have no common outcomes.

**Example**:
Consider a six-sided die roll:
- Event A: Rolling an odd number = {1, 3, 5}
- Event B: Rolling an even number = {2, 4, 6}

Events A and B are mutually exclusive because you cannot roll both an odd and even number in a single roll of a die. If one event happens, the other cannot.

In mathematical terms: 
\[
P(A \cap B) = 0
\]
(where \(A \cap B\) represents the intersection of A and B, or outcomes common to both events, which is empty).

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### 2. Mutually Exhaustive Events:

**Definition**:
Two or more events are **mutually exhaustive** if their union covers all possible outcomes of an experiment. In other words, at least one of the events must occur when the experiment is conducted.

**Example**:
Again, consider rolling a six-sided die:
- Event A: Rolling an odd number = {1, 3, 5}
- Event B: Rolling an even number = {2, 4, 6}

Here, Events A and B are mutually exhaustive because together they cover all possible outcomes of rolling a die (either an odd or even number must be rolled). The union of A and B includes every possible outcome: 
\[
A \cup B = \{1, 2, 3, 4, 5, 6\}
\]
In mutually exhaustive events, the sum of the probabilities is 1:
\[
P(A \cup B) = 1
\]

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### Combined Example:

Imagine you're flipping a coin:
- Event A: Getting a head.
- Event B: Getting a tail.

These events are both **mutually exclusive** (you can't get both heads and tails at the same time) and **mutually exhaustive** (the only possible outcomes are heads or tails, so their union covers all possibilities).

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### Summary:
- **Mutually exclusive**: Events cannot happen together.
- **Mutually exhaustive**: Events together cover all possible outcomes.

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Do you want more details or have any questions?

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### Related Questions:
1. What are other examples of mutually exclusive events in real life?
2. Can two events be mutually exclusive but not exhaustive?
3. What is the probability rule for mutually exclusive events?
4. How do mutually exhaustive events relate to the concept of sample space?
5. Can more than two events be mutually exclusive or exhaustive?

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**Tip**: When dealing with probabilities, always check if events are mutually exclusive or exhaustive. It simplifies problem-solving!

Explain mutually exclusive events and mutually exhaustive events with examples.

Understand the difference between mutually exclusive and mutually exhaustive events in probability theory. This explanation includes definitions, real-world examples, and key formulas for both concepts. Perfect for high school students in Grades 9-12.

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