A player rolls a die and receives the same number of rupees as the number of dots on the face that turns up. What should the player pay for each roll if he wants to make a profit of one rupee per throw of the die in the long run ? Assume that the die has six faces carrying numbers 1 through 6.

Options

A. Rs. 2.50
B. Rs. 2
C. Rs. 3.50
D. Rs. 4

Answer

To solve the problem, we need to determine how much the player should pay for each roll of the die in order to achieve a profit of 1 rupee per throw in the long run.

### Step-by-Step Solution:

**Step 1: Calculate the expected value of the winnings from rolling the die.**
- A die has six faces numbered from 1 to 6.
- The expected value (average winnings) can be calculated using the formula for the average of a uniform distribution:

\[ \text{Expected Value} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} \]

- First, calculate the sum of the numbers on the die:

\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]

- Now, divide by the number of faces (6):

\[ \text{Expected Value} = \frac{21}{6} = 3.5 \]

**Step 2: Determine the amount the player should pay to achieve a profit of 1 rupee.**
- The player wants to make a profit of 1 rupee per throw. This means that the amount he receives from rolling the die minus the amount he pays should equal 1 rupee.

Let \( x \) be the amount the player pays for each roll. The equation can be set up as follows:

\[ \text{Expected Winnings} - x = 1 \]

- Substitute the expected winnings (3.5 rupees) into the equation:

\[ 3.5 - x = 1 \]

**Step 3: Solve for \( x \).**
- Rearranging the equation gives:

\[ x = 3.5 - 1 \]

- Therefore:

\[ x = 2.5 \]

**Conclusion:**
The player should pay **2.5 rupees** for each roll of the die to make a profit of 1 rupee per throw in the long run.