The probability of picking a robot card from Jack's pack is \(\frac{10}{15} = \frac{2}{3}\).

Emma has 7 robot cards, so the total number of cards in Emma's pack is \(x + 4\) because \(x - 3\) cards are aeroplanes, making \(x - 3 + 7 = x + 4\) cards in total.

The probability of picking a robot card from Emma's pack is \(\frac{7}{x+4}\).

The probability that both cards have pictures of robots is \(\frac{2}{3} \times \frac{7}{x+4} = \frac{7}{18}\).

Setting up the equation: \(\frac{2}{3} \times \frac{7}{x+4} = \frac{7}{18}\).

Solving for \(x\):

\(\frac{14}{3(x+4)} = \frac{7}{18}\)

Cross-multiply: \(14 \times 18 = 7 \times 3(x+4)\)

\(252 = 21(x+4)\)

\(252 = 21x + 84\)

\(168 = 21x\)

\(x = 8\)