**Step 1 — Total number of outcomes.**
$$n(S) = 6 \times 6 = 36$$

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**(i) Even sum**

**Step 2 — When is a sum even?**
Sum is even when **both numbers are even** or **both numbers are odd**.

**Step 3 — Count.**
- Both even : $3 \times 3 = 9$ ways (from $\{2,4,6\}$)
- Both odd : $3 \times 3 = 9$ ways (from $\{1,3,5\}$)

$$n(E_1) = 9 + 9 = 18$$

**Step 4 — Probability.**
$$P(\text{even sum}) = \frac{18}{36} = \frac12$$

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**(ii) Even product**

**Step 5 — When is a product odd?**
The product is odd **only** when both numbers are odd.
$$n(\text{odd product}) = 3 \times 3 = 9$$

**Step 6 — Count the even products.**
$$n(E_2) = 36 - 9 = 27$$

**Step 7 — Probability.**
$$P(\text{even product}) = \frac{27}{36} = \frac34$$

$$\boxed{\text{(i) } \frac12 \qquad \text{(ii) } \frac34}$$