A die is tossed thrice. A success is getting 5 or 6 on a toss. The mean and the variance of number of successes

Options

A. `mu = 1,sigma^(2) = 2/3`
B. `mu = 2/3,sigma^(2) = 1`
C. `mu = 2,sigma^(2) = 2/3`
D. `mu = 1,sigma^(2) = 1`

Answer

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

### Step-by-Step Solution:

1. **Identify the Number of Trials (n)**:
- A die is tossed thrice, so the number of trials \( n = 3 \).

**Hint**: Count how many times the experiment (tossing the die) is performed.

2. **Determine the Probability of Success (p)**:
- A success is defined as getting a 5 or a 6 on a toss. There are 2 successful outcomes (5 and 6) out of 6 possible outcomes when tossing a die.
- Therefore, the probability of success \( p = \frac{2}{6} = \frac{1}{3} \).

**Hint**: Calculate the ratio of successful outcomes to total outcomes.

3. **Calculate the Probability of Failure (q)**:
- The probability of failure \( q \) is given by \( q = 1 - p \).
- Thus, \( q = 1 - \frac{1}{3} = \frac{2}{3} \).

**Hint**: Use the complement rule to find the probability of failure.

4. **Calculate the Mean (μ)**:
- The mean of a Bernoulli distribution is calculated using the formula \( \mu = n \times p \).
- Substituting the values, we get:
\[ \mu = 3 \times \frac{1}{3} = 1 \]

**Hint**: Multiply the number of trials by the probability of success.

5. **Calculate the Variance (σ²)**:
- The variance of a Bernoulli distribution is calculated using the formula \( \sigma^2 = n \times p \times q \).
- Substituting the values, we get:
\[ \sigma^2 = 3 \times \frac{1}{3} \times \frac{2}{3} = 1 \times \frac{2}{3} = \frac{2}{3} \]

**Hint**: Use the formula for variance, which incorporates both the probability of success and the probability of failure.

### Final Answer:
- The mean number of successes is \( 1 \).
- The variance of the number of successes is \( \frac{2}{3} \).