A die is rolled thrice . If the event of getting an even number is a success , then the probability of getting atleast two sucessess is

Options

A. `7/8`
B. `1/4`
C. `2/3`
D. `1/2`

Answer

To solve the problem of finding the probability of getting at least two successes when a die is rolled thrice, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Probability of Success**:
- When rolling a die, the even numbers are 2, 4, and 6. Therefore, there are 3 successful outcomes (even numbers) out of 6 possible outcomes.
- The probability of getting an even number (success) is:
\[ P(\text{success}) = \frac{3}{6} = \frac{1}{2} \]

2. **Determine the Probability of Failure**:
- The probability of not getting an even number (failure) is:
\[ P(\text{failure}) = 1 - P(\text{success}) = 1 - \frac{1}{2} = \frac{1}{2} \]

3. **Define the Random Variable**:
- Let \( X \) be the random variable representing the number of successes (even numbers) in 3 rolls of the die. \( X \) follows a binomial distribution with parameters \( n = 3 \) (number of trials) and \( p = \frac{1}{2} \) (probability of success).

4. **Calculate the Probability of Getting at Least 2 Successes**:
- We need to find \( P(X \geq 2) \), which can be calculated as:
\[ P(X \geq 2) = P(X = 2) + P(X = 3) \]

5. **Calculate \( P(X = 2) \)**:
- Using the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
- For \( r = 2 \):
\[ P(X = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{3-2} = 3 \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{8} \]

6. **Calculate \( P(X = 3) \)**:
- For \( r = 3 \):
\[ P(X = 3) = \binom{3}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{3-3} = 1 \cdot \frac{1}{8} \cdot 1 = \frac{1}{8} \]

7. **Combine the Probabilities**:
- Now, we can add the probabilities:
\[ P(X \geq 2) = P(X = 2) + P(X = 3) = \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

### Final Answer:
The probability of getting at least two successes when a die is rolled thrice is:
\[ \frac{1}{2} \]