John is playing a game with two urns that look identical. The first urn has 4 green marbles and 2 blue marbles, and the second urn has 3 green marbles and 3 blue marbles. John will draw a marble from one of the two urns, and then without replacement, he will draw another marble from one of the two urns. He wins a prize if he draws one of each color.

He starts the game and chooses one of the two urns uniformly at random and then draws one marble from it uniformly at random. He checks the color of the marble he drew and sees that it's green. Now he needs to decide which urn to draw his second marble from. He has three choices for drawing the second marble:

1. Draw the next marble from the other urn
2. Draw the next marble from the same urn
3. Choose an urn uniformly at random and draw the next marble from that urn.

Which choice maximizes the probability of John winning the game? Using this choice, what is the variance of the number of blue marbles he ends up with?