Two Blue Gumballs

An arcade has two gumball machines, one left and one right, each filled with blue and yellow gumballs. Each gumball from the left machine is blue with probability $0.3$; each from the right machine is blue with probability $0.6$. You flip a fair coin to pick one machine and draw 2 gumballs from it; both are blue. What is the probability that you were drawing from the left machine? Assume draws are independent (each machine holds so many gumballs that removing one does not change the probabilities). Give an exact fraction.