Answer
**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: {}
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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