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

Options

A. 1
B. 2
C. 5
D. `8/5`

Explanation

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

### Step 1: Identify the outcomes and their probabilities
The die has:
- 1 on 3 faces
- 2 on 2 faces
- 5 on 1 face

The total number of faces on the die is 6.

**Probabilities:**
- Probability of rolling a 1 (P(X=1)): There are 3 faces with 1, so P(X=1) = 3/6 = 1/2.
- Probability of rolling a 2 (P(X=2)): There are 2 faces with 2, so P(X=2) = 2/6 = 1/3.
- Probability of rolling a 5 (P(X=5)): There is 1 face with 5, so P(X=5) = 1/6.

### Step 2: Set up the mean formula
The mean (expected value) of a discrete random variable is calculated using the formula:
\[ \text{Mean} = E(X) = \sum (X \cdot P(X)) \]

### Step 3: Calculate the mean
Now, we will calculate the mean using the probabilities we found:

\[ E(X) = (1 \cdot P(X=1)) + (2 \cdot P(X=2)) + (5 \cdot P(X=5)) \]

Substituting the probabilities:

\[ E(X) = (1 \cdot \frac{1}{2}) + (2 \cdot \frac{1}{3}) + (5 \cdot \frac{1}{6}) \]

Calculating each term:

1. \( 1 \cdot \frac{1}{2} = \frac{1}{2} \)
2. \( 2 \cdot \frac{1}{3} = \frac{2}{3} \)
3. \( 5 \cdot \frac{1}{6} = \frac{5}{6} \)

### Step 4: Find a common denominator and sum the fractions
The common denominator for \( \frac{1}{2}, \frac{2}{3}, \frac{5}{6} \) is 6.

Converting each fraction:
- \( \frac{1}{2} = \frac{3}{6} \)
- \( \frac{2}{3} = \frac{4}{6} \)
- \( \frac{5}{6} = \frac{5}{6} \)

Now, adding them together:

\[ E(X) = \frac{3}{6} + \frac{4}{6} + \frac{5}{6} = \frac{3 + 4 + 5}{6} = \frac{12}{6} = 2 \]

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

---