20) B

Since the balls are arranged in a way to give the smallest possibility for a player to win, the first bag must have 2 red balls and 1 yellow ball and 1 blue ball. The second bag must have 2 red balls and 2 yellow balls.

Probability of choosing either bag = 1/2

Let us say a player picked bag 1.
The only way to win is to pick 2 red balls.
Probability of picking a red ball = 2/4 = 1/2
In order to win, the player must pick a second red ball.
Probability of picking a second red ball = 1/3
Probability of picking two same coloured balls from Bag 1 = 1/2 x 1/2 x 1/3 = 1/12

Let us say a player picked bag 2.
Only way to win is to either pick 2 red balls or 2 yellow balls.

1st ball
Probability of picking a first red ball = 2/4 = 1/2
Probability of picking a first yellow ball = 2/4 = 1/2

2nd ball
If the player picked a red ball first, they must pick a red ball again if they want to win.
Probability of picking a second red ball = 1/3

If the player picked a yellow ball first, they must pick a yellow ball again if they want to win.
Probability of picking a second yellow ball = 1/3

Probability of picking two same coloured balls from Bag 2 = 2( 1/2 x 1/2 x 1/3 ) = 1/12 x 2 = 2/12

There are three possibilities to win:
2 red balls from bag 1 = Probability of 1/12
2 red balls from bag 2 = Probability of 1/12
2 yellow balls from bag 2 = Probability of 1/12

Probability of winning = (1/12 x 3) = 3/12 = 1/4