(i) The probability that Ben becomes the champion after playing exactly 2 games is when Ben wins both games. The probability of Ben winning a game is 0.7. Therefore, the probability is:

\(P(B \text{ champ}) = 0.7 \times 0.7 = 0.49\)

(ii) The probability that Ben becomes the champion can occur in the following sequences: WW, WLW, LWW.

\(P(B \text{ champ}) = P(WW) + P(WLW) + P(LWW)\)

\(= (0.7 \times 0.7) + (0.7 \times 0.3 \times 0.7) + (0.3 \times 0.7 \times 0.7)\)

\(= 0.49 + 0.147 + 0.147\)

\(= 0.784\)

(iii) Given that Tom becomes the champion, we need to find the probability that he won the 2nd game. This can occur in the sequences: TT or TBT.

\(P(T2 \cap T) = (0.3 \times 0.3) + (0.7 \times 0.3 \times 0.3) = 0.216\)

The probability that Tom becomes the champion is \(1 - 0.784 = 0.216\).

Thus, the probability is:

\(P(T2 | T) = \frac{P(T2 \cap T)}{P(T)} = \frac{0.216}{0.216} = 0.708\)