A die is thrown thrice. If getting a four is considered a success, then the mean and variance of the distribution of the number of successes are

Options

A. `1/2, 5/12`
B. `1/6, 5/12`
C. `5/6, 1/2`
D. `1/6, 1/2`

Answer

To solve the problem of finding the mean and variance of the distribution of the number of successes when a die is thrown thrice, we can follow these steps:

### Step 1: Define the Probability of Success
When a die is thrown, the probability of getting a 4 (considered a success) is:
\[ p = \frac{1}{6} \]

### Step 2: Define the Probability of Failure
The probability of not getting a 4 (considered a failure) is:
\[ q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 3: Determine the Number of Trials
Since the die is thrown three times, the number of trials \( n \) is:
\[ n = 3 \]

### Step 4: Calculate the Mean
The mean \( \mu \) of a Bernoulli distribution is given by the formula:
\[ \mu = n \cdot p \]
Substituting the values:
\[ \mu = 3 \cdot \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 5: Calculate the Variance
The variance \( \sigma^2 \) of a Bernoulli distribution is given by the formula:
\[ \sigma^2 = n \cdot p \cdot q \]
Substituting the values:
\[ \sigma^2 = 3 \cdot \frac{1}{6} \cdot \frac{5}{6} = 3 \cdot \frac{5}{36} = \frac{15}{36} = \frac{5}{12} \]

### Conclusion
Thus, the mean and variance of the distribution of the number of successes are:
- Mean \( \mu = \frac{1}{2} \)
- Variance \( \sigma^2 = \frac{5}{12} \)

### Final Answer
Mean: \( \frac{1}{2} \)
Variance: \( \frac{5}{12} \)

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