First, calculate the probability of event R (product is 12). The possible pairs are (3, 4), (4, 3), (2, 6), and (6, 2). Thus,

\(P(R) = \frac{4}{36} = \frac{1}{9}\).

Next, calculate the probability of event T (one number is odd, one is even). There are 18 such pairs out of 36 total possibilities, so

\(P(T) = \frac{18}{36} = \frac{1}{2}\).

Now, calculate the probability of both events occurring together, \(P(R \cap T)\). The pairs (3, 4) and (4, 3) satisfy both conditions, so

\(P(R \cap T) = \frac{2}{36} = \frac{1}{18}\).

Check for independence by verifying if \(P(R) \times P(T) = P(R \cap T)\):

\(\frac{1}{9} \times \frac{1}{2} = \frac{1}{18}\).

Since \(P(R) \times P(T) = P(R \cap T)\), the events are independent.