The mean of the numbers obtained on throwing a die having written 1 one three faces, 2 on two faces and 5 on one face is

Options

A. 1
B. 2
C. 5
D. `(8)/(3)`

Answer

To find the mean of the numbers obtained on throwing a die with specific values on its faces, we can follow these steps:

### Step 1: Identify the values and their frequencies
The die has:
- The number 1 on 3 faces
- The number 2 on 2 faces
- The number 5 on 1 face

### Step 2: Determine the total number of faces
The total number of faces on the die is 6.

### Step 3: Calculate the probabilities for each number
- Probability of rolling a 1: \( P(1) = \frac{3}{6} = \frac{1}{2} \)
- Probability of rolling a 2: \( P(2) = \frac{2}{6} = \frac{1}{3} \)
- Probability of rolling a 5: \( P(5) = \frac{1}{6} \)

### Step 4: Use the formula for mean
The mean \( \mu \) can be calculated using the formula:
\[ \mu = \sum (x \cdot P(x)) \]
Where \( x \) is the value and \( P(x) \) is the probability of that value.

### Step 5: Substitute the values into the formula
Now we substitute the values and their probabilities:
\[ \mu = (1 \cdot \frac{1}{2}) + (2 \cdot \frac{1}{3}) + (5 \cdot \frac{1}{6}) \]

### Step 6: Calculate each term
- For \( 1 \cdot \frac{1}{2} = \frac{1}{2} \)
- For \( 2 \cdot \frac{1}{3} = \frac{2}{3} \)
- For \( 5 \cdot \frac{1}{6} = \frac{5}{6} \)

### Step 7: Find a common denominator and sum the fractions
The least common multiple (LCM) of the denominators (2, 3, and 6) is 6. We convert each fraction:
- \( \frac{1}{2} = \frac{3}{6} \)
- \( \frac{2}{3} = \frac{4}{6} \)
- \( \frac{5}{6} = \frac{5}{6} \)

Now we can sum them up:
\[ \mu = \frac{3}{6} + \frac{4}{6} + \frac{5}{6} = \frac{3 + 4 + 5}{6} = \frac{12}{6} = 2 \]

### Final Answer
The mean of the numbers obtained on throwing the die is **2**.

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