(i) A die is thrown 5 times. If getting an 'odd number' is success, find the probability of getting at least 4 successes.
(ii) A die is thrown 6 times. If getting an 'odd (even) number' is a success, what is the probability of :
(I) 5 successes (II) at least 5 successes (III) at most 5 successes (IV) no success ?
(iii) A die is thrown 10 times. If getting an even number is considered a success, find the probability of at least 9 successes.

Answer

To solve the given problems step by step, we will use the concepts of probability, specifically the binomial probability formula. The binomial probability formula is given by:

\[ P(X = r) = nCr \cdot p^r \cdot q^{n-r} \]

where:
- \( n \) = number of trials
- \( r \) = number of successes
- \( p \) = probability of success
- \( q \) = probability of failure (where \( q = 1 - p \))
- \( nCr \) = binomial coefficient, which is calculated as \( \frac{n!}{r!(n-r)!} \)

### Part (i)
A die is thrown 5 times. If getting an 'odd number' is success, find the probability of getting at least 4 successes.

1. **Identify the parameters:**
- Number of trials, \( n = 5 \)
- Probability of success (getting an odd number), \( p = \frac{3}{6} = \frac{1}{2} \)
- Probability of failure (getting an even number), \( q = 1 - p = \frac{1}{2} \)

2. **Calculate the probability of getting at least 4 successes:**
- This means we need to find \( P(X \geq 4) = P(X = 4) + P(X = 5) \).

3. **Calculate \( P(X = 4) \):**
\[ P(X = 4) = 5C4 \cdot \left(\frac{1}{2}\right)^4 \cdot \left(\frac{1}{2}\right)^{5-4} = 5 \cdot \left(\frac{1}{2}\right)^5 = 5 \cdot \frac{1}{32} = \frac{5}{32} \]

4. **Calculate \( P(X = 5) \):**
\[ P(X = 5) = 5C5 \cdot \left(\frac{1}{2}\right)^5 \cdot \left(\frac{1}{2}\right)^{5-5} = 1 \cdot \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]

5. **Combine the probabilities:**
\[ P(X \geq 4) = P(X = 4) + P(X = 5) = \frac{5}{32} + \frac{1}{32} = \frac{6}{32} = \frac{3}{16} \]

### Part (ii)
A die is thrown 6 times. If getting an 'odd (even) number' is a success, find the probabilities for the following:

**(I) Probability of 5 successes:**
1. **Parameters:**
- \( n = 6 \), \( p = \frac{1}{2} \), \( q = \frac{1}{2} \)

2. **Calculate \( P(X = 5) \):**
\[ P(X = 5) = 6C5 \cdot \left(\frac{1}{2}\right)^5 \cdot \left(\frac{1}{2}\right)^{6-5} = 6 \cdot \left(\frac{1}{2}\right)^6 = 6 \cdot \frac{1}{64} = \frac{6}{64} = \frac{3}{32} \]

**(II) Probability of at least 5 successes:**
1. **Calculate \( P(X \geq 5) = P(X = 5) + P(X = 6) \)**

2. **Calculate \( P(X = 6) \):**
\[ P(X = 6) = 6C6 \cdot \left(\frac{1}{2}\right)^6 = 1 \cdot \frac{1}{64} = \frac{1}{64} \]

3. **Combine the probabilities:**
\[ P(X \geq 5) = P(X = 5) + P(X = 6) = \frac{3}{32} + \frac{1}{64} = \frac{6}{64} + \frac{1}{64} = \frac{7}{64} \]

**(III) Probability of at most 5 successes:**
1. **Calculate \( P(X \leq 5) = 1 - P(X = 6) \)**
\[ P(X \leq 5) = 1 - \frac{1}{64} = \frac{63}{64} \]

**(IV) Probability of no successes:**
1. **Calculate \( P(X = 0) \):**
\[ P(X = 0) = 6C0 \cdot \left(\frac{1}{2}\right)^0 \cdot \left(\frac{1}{2}\right)^6 = 1 \cdot 1 \cdot \frac{1}{64} = \frac{1}{64} \]

### Part (iii)
A die is thrown 10 times. If getting an even number is considered a success, find the probability of at least 9 successes.

1. **Identify the parameters:**
- \( n = 10 \), \( p = \frac{1}{2} \), \( q = \frac{1}{2} \)

2. **Calculate \( P(X \geq 9) = P(X = 9) + P(X = 10) \)**

3. **Calculate \( P(X = 9) \):**
\[ P(X = 9) = 10C9 \cdot \left(\frac{1}{2}\right)^9 \cdot \left(\frac{1}{2}\right)^{10-9} = 10 \cdot \left(\frac{1}{2}\right)^{10} = 10 \cdot \frac{1}{1024} = \frac{10}{1024} = \frac{5}{512} \]

4. **Calculate \( P(X = 10) \):**
\[ P(X = 10) = 10C10 \cdot \left(\frac{1}{2}\right)^{10} = 1 \cdot \frac{1}{1024} = \frac{1}{1024} \]

5. **Combine the probabilities:**
\[ P(X \geq 9) = P(X = 9) + P(X = 10) = \frac{5}{512} + \frac{1}{1024} = \frac{10}{1024} + \frac{1}{1024} = \frac{11}{1024} \]

### Summary of Answers:
- (i) Probability of at least 4 successes: \( \frac{3}{16} \)
- (ii)
- (I) Probability of 5 successes: \( \frac{3}{32} \)
- (II) Probability of at least 5 successes: \( \frac{7}{64} \)
- (III) Probability of at most 5 successes: \( \frac{63}{64} \)
- (IV) Probability of no successes: \( \frac{1}{64} \)
- (iii) Probability of at least 9 successes: \( \frac{11}{1024} \)