A die is thrown 6 times . If getting an even number is a success find the probability of getting .
(i) exactly 5 successes (ii) at least 5 successes
(iii) at most 5 successes .

Explanation

To solve the problem, we will use the binomial probability formula. The probability of success (getting an even number) when rolling a die is \( P = \frac{3}{6} = \frac{1}{2} \) (since the even numbers are 2, 4, and 6). The probability of failure (getting an odd number) is \( Q = 1 - P = \frac{1}{2} \).

Given:
- Number of trials (N) = 6
- Probability of success (P) = \( \frac{1}{2} \)
- Probability of failure (Q) = \( \frac{1}{2} \)

The binomial probability formula is given by:
\[ P(X = r) = \binom{N}{r} P^r Q^{N-r} \]

### (i) Probability of getting exactly 5 successes

1. **Identify the values**:
- \( N = 6 \)
- \( r = 5 \)
- \( P = \frac{1}{2} \)
- \( Q = \frac{1}{2} \)

2. **Use the binomial formula**:
\[ P(X = 5) = \binom{6}{5} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{6-5} \]

3. **Calculate \( \binom{6}{5} \)**:
\[ \binom{6}{5} = 6 \]

4. **Calculate the probability**:
\[ P(X = 5) = 6 \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^1 = 6 \left(\frac{1}{2}\right)^6 = 6 \cdot \frac{1}{64} = \frac{6}{64} = \frac{3}{32} \]

### (ii) Probability of getting at least 5 successes

1. **At least 5 successes means**: \( P(X \geq 5) = P(X = 5) + P(X = 6) \)

2. **Calculate \( P(X = 6) \)**:
\[ P(X = 6) = \binom{6}{6} \left(\frac{1}{2}\right)^6 \left(\frac{1}{2}\right)^{0} = 1 \cdot \left(\frac{1}{2}\right)^6 = \frac{1}{64} \]

3. **Combine the probabilities**:
\[ P(X \geq 5) = P(X = 5) + P(X = 6) = \frac{3}{32} + \frac{1}{64} \]

4. **Convert \( \frac{3}{32} \) to a common denominator**:
\[ \frac{3}{32} = \frac{6}{64} \]

5. **Final calculation**:
\[ P(X \geq 5) = \frac{6}{64} + \frac{1}{64} = \frac{7}{64} \]

### (iii) Probability of getting at most 5 successes

1. **At most 5 successes means**: \( P(X \leq 5) = 1 - P(X = 6) \)

2. **Using the previously calculated value**:
\[ P(X \leq 5) = 1 - P(X = 6) = 1 - \frac{1}{64} = \frac{64 - 1}{64} = \frac{63}{64} \]

### Summary of Results
- (i) Probability of exactly 5 successes: \( \frac{3}{32} \)
- (ii) Probability of at least 5 successes: \( \frac{7}{64} \)
- (iii) Probability of at most 5 successes: \( \frac{63}{64} \)