To solve the problem, we need to calculate the expected income per throw of a biased die, where the outcomes and their associated rewards are given. Let's break it down step by step.
### Step 1: Define the probabilities
Given that the number 5 appears 5 times as frequently as any other number, we can define the probabilities as follows:
- Let the probability of rolling any number other than 5 (1, 2, 3, 4, or 6) be \( m \).
- Therefore, the probability of rolling a 5 will be \( 5m \).
Since there are 6 faces on a die, the total probability must equal 1:
\[
m + m + m + m + m + 5m = 1
\]
This simplifies to:
\[
10m = 1
\]
### Step 2: Solve for \( m \)
Now, we can solve for \( m \):
\[
m = \frac{1}{10}
\]
### Step 3: Calculate individual probabilities
Now we can determine the probabilities for each outcome:
- Probability of rolling a 1, 2, 3, 4, or 6: \( m = \frac{1}{10} \)
- Probability of rolling a 5: \( 5m = 5 \times \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \)
### Step 4: Calculate expected income
Next, we calculate the expected income based on the rewards:
- If the number rolled is 5, the income is 5 Rupees.
- If the number rolled is 1, 2, 3, 4, or 6, the income is 8 Rupees.
The expected income \( E \) can be calculated as follows:
\[
E = (P(5) \times \text{Income from 5}) + (P(1) \times \text{Income from 1}) + (P(2) \times \text{Income from 2}) + (P(3) \times \text{Income from 3}) + (P(4) \times \text{Income from 4}) + (P(6) \times \text{Income from 6})
\]
Substituting the probabilities and incomes:
\[
E = \left(\frac{1}{2} \times 5\right) + \left(\frac{1}{10} \times 8\right) + \left(\frac{1}{10} \times 8\right) + \left(\frac{1}{10} \times 8\right) + \left(\frac{1}{10} \times 8\right) + \left(\frac{1}{10} \times 8\right)
\]
### Step 5: Simplify the expression
Calculating each term:
- For rolling a 5: \( \frac{1}{2} \times 5 = 2.5 \)
- For rolling 1, 2, 3, 4, or 6: Each contributes \( \frac{1}{10} \times 8 = 0.8 \)
Since there are 5 such outcomes (1, 2, 3, 4, 6):
\[
E = 2.5 + 5 \times 0.8 = 2.5 + 4 = 6.5
\]
### Final Answer
The expected income per throw is **6.5 Rupees**.
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