Step 1: Define Sample Space and Probabilities

For a single die roll, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. The total number of outcomes is $n(S) = 6$. The probability of any single outcome is $1/6$.

Step 2: Calculate Conditional Probabilities for (i)

We use the formula $P(E|F) = \frac{P(E \cap F)}{P(F)}$. Given $A=\{1,2,5\}$ and $C=\{2,3,4,5\}$, the intersection $A \cap C = \{2, 5\}$. Thus, $n(A \cap C) = 2$.

$$P(A|C) = \frac{n(A \cap C)}{n(C)} = \frac{2}{4} = \frac{1}{2}$$

$$P(C|A) = \frac{n(C \cap A)}{n(A)} = \frac{2}{3}$$

Step 3: Calculate Conditional Probabilities for (ii)

First, find $A \cap B = \{5\}$ and $A \cup B = \{1, 2, 3, 5\}$.

For $P(A \cap B | C)$: The intersection $(A \cap B) \cap C = \{5\}$. Thus, $n((A \cap B) \cap C) = 1$.

$$P(A \cap B | C) = \frac{1}{4}$$

For $P(A \cup B | C)$: The intersection $(A \cup B) \cap C = \{2, 3, 5\}$. Thus, $n((A \cup B) \cap C) = 3$.

$$P(A \cup B | C) = \frac{3}{4}$$