A person plays a game of tossing a coin thrice. For each head, he is given Rs 2 by the organiser of the game and for each tail, he has to give Rs. 1.50 to the organise. Let X denote the amount gained or lost by the person . Thus write the Range of Distribution.

Options

A. {-1, 2 . 50, - 4.50, 6}
B. {-2, 4.50, -3.50, 6}
C. {-3, 4.50, -3.50, 8}
D. {-4, 6.50, -3.50, 6}

Answer

To solve the problem, we need to determine the amount gained or lost by a person who tosses a coin three times and receives Rs 2 for each head and pays Rs 1.50 for each tail. We will denote the amount gained or lost by the person as \( X \).

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( H \) be the number of heads.
- Let \( T \) be the number of tails.
- Since the coin is tossed three times, we have the relationship:
\[ H + T = 3 \]

2. **Calculate the Amount Gained or Lost**:
- The amount gained (for heads) is \( 2H \).
- The amount lost (for tails) is \( 1.5T \).
- Therefore, the total amount \( X \) can be expressed as:
\[ X = 2H - 1.5T \]

3. **Identify Possible Outcomes**:
- The possible outcomes for the number of heads (H) when tossing the coin three times can be 0, 1, 2, or 3. Correspondingly, the number of tails (T) will be 3, 2, 1, or 0 respectively.

4. **Calculate \( X \) for Each Case**:
- **Case 1**: \( H = 3 \), \( T = 0 \)
\[ X = 2(3) - 1.5(0) = 6 \]
- **Case 2**: \( H = 2 \), \( T = 1 \)
\[ X = 2(2) - 1.5(1) = 4 - 1.5 = 2.5 \]
- **Case 3**: \( H = 1 \), \( T = 2 \)
\[ X = 2(1) - 1.5(2) = 2 - 3 = -1 \]
- **Case 4**: \( H = 0 \), \( T = 3 \)
\[ X = 2(0) - 1.5(3) = 0 - 4.5 = -4.5 \]

5. **Determine the Range of Distribution**:
- From the calculations above, the possible values of \( X \) are:
- \( 6 \) (3 heads)
- \( 2.5 \) (2 heads, 1 tail)
- \( -1 \) (1 head, 2 tails)
- \( -4.5 \) (0 heads, 3 tails)
- Thus, the range of distribution for \( X \) is:
\[ \text{Range} = \{-4.5, -1, 2.5, 6\} \]

### Final Answer:
The range of distribution for the amount gained or lost by the person is:
\[ \{-4.5, -1, 2.5, 6\} \]