To find the probability that Ronnie wins, we consider the scenarios where Ronnie draws the first yellow ball.

1. Probability that Julie draws a green ball first and Ronnie draws a yellow ball second:

\(P(GY) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56}\)

2. Probability that both Julie and Ronnie draw green balls first, and Julie draws a green ball again, then Ronnie draws a yellow ball:

\(P(GGGY) = \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{3}{5} = \frac{3}{28}\)

3. Probability that the first five draws are green balls, and Ronnie draws a yellow ball on his third turn:

\(P(GGGGGY) = \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{2}{5} \times \frac{1}{4} \times \frac{3}{3} = \frac{1}{56}\)

Adding these probabilities gives the total probability that Ronnie wins:

\(P(R \text{ wins}) = \frac{15}{56} + \frac{3}{28} + \frac{1}{56} = \frac{11}{28}\)