A person who tosses an unbiased coin gains two points for turning up a head and loses one point for a tail. If three coins are tossed and the total score X is observed, then the range of X is

Options

A. {0, 3, 6}
B. {-3, 0, 3}
C. {-3, 0, 3, 6}
D. {-3, 3, 6}

Answer

To solve the problem, we need to determine the total score \( X \) when tossing three unbiased coins, where gaining points for heads and losing points for tails is defined as follows:

- Gain 2 points for each head (H)
- Lose 1 point for each tail (T)

### Step-by-Step Solution:

1. **Identify Possible Outcomes**:
When tossing three coins, the possible outcomes in terms of heads (H) and tails (T) can be:
- 3 heads (HHH)
- 2 heads and 1 tail (HHT, HTH, THH)
- 1 head and 2 tails (HTT, THT, TTH)
- 3 tails (TTT)

2. **Calculate Scores for Each Outcome**:
- **3 Heads (HHH)**:
\[ X = 3 \times 2 = 6 \text{ points} \]
- **2 Heads and 1 Tail (HHT, HTH, THH)**:
\[ X = 2 \times 2 + 1 \times (-1) = 4 - 1 = 3 \text{ points} \]
- **1 Head and 2 Tails (HTT, THT, TTH)**:
\[ X = 1 \times 2 + 2 \times (-1) = 2 - 2 = 0 \text{ points} \]
- **3 Tails (TTT)**:
\[ X = 0 \times 2 + 3 \times (-1) = 0 - 3 = -3 \text{ points} \]

3. **List All Possible Scores**:
From the calculations above, the possible scores \( X \) are:
- \( 6 \) (from 3 heads)
- \( 3 \) (from 2 heads and 1 tail)
- \( 0 \) (from 1 head and 2 tails)
- \( -3 \) (from 3 tails)

4. **Determine the Range of \( X \)**:
The scores we found are \( -3, 0, 3, 6 \). The range of \( X \) is from the minimum score to the maximum score.

5. **Final Result**:
Thus, the range of \( X \) is:
\[ \text{Range of } X = [-3, 6] \]