Гry These: Show your answers. ( $$ 1 / 4 $$ sheet of paper) A die is rolled. . List all the outcomes 2. List all the sample space 3. List all the events

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Certainly! Let's tackle each part of the problem step by step.

### 1. List All the Outcomes

When you roll a standard six-sided die, each face represents a possible outcome. The outcomes are the numbers that can appear on the top face of the die after a roll.

**Outcomes:**
- **1**
- **2**
- **3**
- **4**
- **5**
- **6**

### 2. List the Sample Space

The **sample space** is a comprehensive set that includes all possible outcomes of a random experiment. In this case, the experiment is rolling a die.

**Sample Space (S):**
$$ S = \{1, 2, 3, 4, 5, 6\} $$

### 3. List All the Events

An **event** is any subset of the sample space. This means it can consist of one outcome, multiple outcomes, or even no outcomes at all. Since the sample space has 6 outcomes, there are $$2^6 = 64$$ possible events. However, it's impractical to list all 64 events here. Instead, I'll provide examples of different types of events you might consider:

#### a. **Single Outcome Events (Simple Events):**
These events consist of exactly one outcome.

- **Event A:** Rolling a **3**
  $$ A = \{3\} $$
  
- **Event B:** Rolling a **5**
  $$ B = \{5\} $$

#### b. **Multiple Outcome Events:**
These events consist of two or more outcomes.

- **Event C:** Rolling an **even number**
  $$ C = \{2, 4, 6\} $$
  
- **Event D:** Rolling an **odd number**
  $$ D = \{1, 3, 5\} $$
  
- **Event E:** Rolling a number **greater than 4**
  $$ E = \{5, 6\} $$
  
- **Event F:** Rolling a number **less than or equal to 3**
  $$ F = \{1, 2, 3\} $$

#### c. **Complementary Events:**
These are pairs of events where one event is the complement of the other.

- **Event G:** Rolling a **number that is not 6**
  $$ G = \{1, 2, 3, 4, 5\} $$
  
- **Event H:** Rolling a **6**
  $$ H = \{6\} $$
  
*(Here, Event G is the complement of Event H, and together they cover the entire sample space.)*

#### d. **Impossible and Certain Events:**
- **Impossible Event:** An event that cannot occur.
  $$ \text{Event I} = \{\} $$
  
- **Certain Event:** An event that is always true.
  $$ \text{Event J} = \{1, 2, 3, 4, 5, 6\} $$

### Summary

- **Outcomes:** 1, 2, 3, 4, 5, 6
- **Sample Space:** $$ S = \{1, 2, 3, 4, 5, 6\} $$
- **Events:** Any subset of the sample space, including examples like rolling an even number (\{2, 4, 6\}), rolling a 3 (\{3\}), rolling greater than 4 (\{5, 6\}), etc.

Understanding these basics will help you analyze probabilities and comprehend how different events relate to one another within the sample space.
**Outcomes:** 1, 2, 3, 4, 5, 6 **Sample Space:** {1, 2, 3, 4, 5, 6} **Events:** - Rolling a 3: {3} - Rolling an even number: {2, 4, 6} - Rolling a number greater than 4: {5, 6} - Rolling a number less than or equal to 3: {1, 2, 3} - Rolling a 6: {6} - Rolling a number not 6: {1, 2, 3, 4, 5} - Rolling a certain event: {1, 2, 3, 4, 5, 6} - Rolling an impossible event: {}

Гry These: Show your answers. ( \( 1 / 4 \) sheet of paper) A die is rolled. . List all the outcomes 2. List all the sample space 3. List all the events

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