Example 3 A coin is tossed and a die is rolled . The outcome of the coin is recorded " 1 " when it shows head and " 0 " when it shows a tail . The random variable R gives the sum of the outcomes of the coin and the die . Provide Sample Space , Range Space , Probability Distribution Table , Mean (write up an interpretation for mean. At least 2 sentences.)

### **Solution: Coin Toss & Die Roll Probability Analysis**  

We are given a **coin toss** and a **die roll**. The outcome of the coin is recorded as:  
- **1** when it lands on **heads**  
- **0** when it lands on **tails**  

The random variable \( R \) represents the **sum** of the coin's outcome and the die's outcome.  

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## **Step 1: Sample Space**  

The sample space consists of all possible outcomes of the coin toss and die roll.  
Let’s list them:

| Coin Outcome | Die Outcome | \( R = \) (Coin Outcome + Die Outcome) |
|-------------|------------|----------------------------------|
| 0 (Tails)   | 1          | 1                                |
| 0 (Tails)   | 2          | 2                                |
| 0 (Tails)   | 3          | 3                                |
| 0 (Tails)   | 4          | 4                                |
| 0 (Tails)   | 5          | 5                                |
| 0 (Tails)   | 6          | 6                                |
| 1 (Heads)   | 1          | 2                                |
| 1 (Heads)   | 2          | 3                                |
| 1 (Heads)   | 3          | 4                                |
| 1 (Heads)   | 4          | 5                                |
| 1 (Heads)   | 5          | 6                                |
| 1 (Heads)   | 6          | 7                                |

So the **sample space** (all possible outcomes of \( R \)) is:  
\[
S = \{ (0

Example 3 A coin is tossed and a die is rolled. The outcome of the coin is recorded '1' when it shows head and '0' when it shows a tail. The random variable R gives the sum of the outcomes of the coin and the die. Provide Sample Space, Range Space, Probability Distribution Table, Mean (write up an interpretation for mean. At least 2 sentences.)

Explore the probability analysis of a coin toss and die roll scenario, where the outcomes are recorded as 0 for tails and 1 for heads, and the sum of these outcomes defines a random variable. Learn how to calculate the sample space, range space, and mean for this problem, with a clear interpretation of the mean. Suitable for students in Grades 6-8.

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Solution: Coin Toss & Die Roll Probability Analysis

Step 1: Sample Space

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Coin Outcome	Die Outcome	$R =$ (Coin Outcome + Die Outcome)
0 (Tails)	1	1
0 (Tails)	2	2
0 (Tails)	3	3
0 (Tails)	4	4
0 (Tails)	5	5
0 (Tails)	6	6
1 (Heads)	1	2
1 (Heads)	2	3
1 (Heads)	3	4
1 (Heads)	4	5
1 (Heads)	5	6
1 (Heads)	6	7