When a die is rolled twice, if the event of getting an even number is denoted by a success and the number of successes as a random variable, then distribution and mean of the variate are

Options

A. `{:(0,1,2),(1//4,1//2,1//4):}, mu = 1`
B. `{:(0,1,2),(1//3,1//2,1//4):}, mu = 2`
C. `{:(0,1,2),(1//2,1//3,1//4):}, mu = 3`
D. `{:(0,1,2,3),(1//4,1//4,1//4,1//4):}, mu = 1`

Answer

To solve the problem of finding the distribution and mean of the random variable \(X\) (the number of successes when a die is rolled twice, where a success is defined as rolling an even number), we will follow these steps:

### Step 1: Define the Random Variable
Let \(X\) be the random variable representing the number of even numbers rolled when a die is rolled twice. The possible values for \(X\) are 0, 1, or 2.

### Step 2: Determine the Probability of Rolling an Even Number
When rolling a die, the even numbers are 2, 4, and 6. Therefore, the probability of rolling an even number (success) is:
\[ P(\text{Even}) = \frac{3}{6} = \frac{1}{2} \]
Similarly, the probability of rolling an odd number (failure) is also:
\[ P(\text{Odd}) = \frac{3}{6} = \frac{1}{2} \]

### Step 3: Calculate the Probability Distribution
Now we will calculate the probabilities for each possible value of \(X\):

1. **Probability that \(X = 0\)** (no even numbers):
- Both rolls are odd.
\[ P(X = 0) = P(\text{Odd}) \times P(\text{Odd}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

2. **Probability that \(X = 1\)** (one even number):
- This can happen in two ways: either the first roll is even and the second is odd, or the first roll is odd and the second is even.
\[ P(X = 1) = P(\text{Even}) \times P(\text{Odd}) + P(\text{Odd}) \times P(\text{Even}) = \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \]

3. **Probability that \(X = 2\)** (two even numbers):
- Both rolls are even.
\[ P(X = 2) = P(\text{Even}) \times P(\text{Even}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

### Step 4: Summarize the Probability Distribution
We can summarize the probabilities in a table:

| \(X\) | Probability \(P(X)\) |
|-------|----------------------|
| 0 | \(\frac{1}{4}\) |
| 1 | \(\frac{1}{2}\) |
| 2 | \(\frac{1}{4}\) |

### Step 5: Calculate the Mean of the Random Variable
The mean (or expected value) of \(X\) is calculated using the formula:
\[ E(X) = \sum (X \cdot P(X)) \]
Calculating each term:
\[ E(X) = 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} \]
\[ E(X) = 0 + \frac{1}{2} + \frac{2}{4} = \frac{1}{2} + \frac{1}{2} = 1 \]

### Final Result
The distribution of the random variable \(X\) is given by the table above, and the mean of \(X\) is:
\[ E(X) = 1 \]