Throwing a biased die, a person will get 5 Rupees if the throws the number 5 and will get 8 Rupees for any other number, then the expected income (in Rupees) per throw is (it is given that the number 5 will appear 5 times as frequently as any other number)

Options

A. 6.5
B. 7
C. 5
D. 5.5

Answer

To solve the problem of calculating the expected income per throw of a biased die, we will follow these steps:

### Step 1: Define the probabilities
We know that the number 5 appears 5 times as frequently as any other number on the die. Let's denote the probability of rolling any number other than 5 as \( M \). Since there are 5 other numbers (1, 2, 3, 4, and 6), the total probability for these numbers is \( 5M \). The probability of rolling a 5 is \( 5M \).

Since the total probability must equal 1, we can set up the equation:
\[ 5M + 5M = 1 \]
This simplifies to:
\[ 10M = 1 \]
Thus, we find:
\[ M = \frac{1}{10} \]

### Step 2: Calculate the probabilities
Now we can calculate the probabilities for each outcome:
- Probability of rolling a 5:
\[ P(5) = 5M = 5 \times \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \]
- Probability of rolling any other number (1, 2, 3, 4, 6):
\[ P(1) = P(2) = P(3) = P(4) = P(6) = M = \frac{1}{10} \]

### Step 3: Calculate the expected income
The expected income \( E \) can be calculated using the formula:
\[ E = (P(5) \times \text{Income from 5}) + \sum (P(\text{other numbers}) \times \text{Income from other numbers}) \]
Substituting the values:
- Income from rolling a 5: 5 Rupees
- Income from rolling any other number: 8 Rupees

Thus, we have:
\[ E = \left(\frac{1}{2} \times 5\right) + \left(5 \times \frac{1}{10} \times 8\right) \]
Calculating each part:
- For rolling a 5:
\[ \frac{1}{2} \times 5 = 2.5 \]
- For rolling any other number:
\[ 5 \times \frac{1}{10} \times 8 = \frac{40}{10} = 4 \]

Now, adding these two results together:
\[ E = 2.5 + 4 = 6.5 \]

### Conclusion
The expected income per throw is:
\[ \boxed{6.5} \text{ Rupees} \]