Let \(W_1\) be the event that John wins his first game and \(W_2\) be the event that he wins his second game.

The probability that John wins his second game, \(P(W_2)\), can be calculated as:

\(P(W_2) = P(W_1 \cap W_2) + P(L_1 \cap W_2)\)

\(= 0.3 \times 0.6 + 0.7 \times 0.15\)

\(= 0.18 + 0.105\)

\(= 0.285\)

We need to find \(P(W_1 | W_2)\), the probability that John won his first game given that he won his second game. This is given by:

\(P(W_1 | W_2) = \frac{P(W_1 \cap W_2)}{P(W_2)}\)

\(= \frac{0.18}{0.285}\)

\(= 0.632\)

Thus, the probability that John won his first game given that he won his second game is \(0.632\) or \(\frac{12}{19}\).