Intuition for probability of drawing first ball = probability of drawing second ball

Solution 1:

Let us just keep drawing balls and line them up as we do so; thus forming a line of $r$ red and $g$ green balls in order of withdrawal.

I now point to any ball in the line.   What is the probability that it is red?

Should it matter at all where in the line I have pointed?   The first?   The second?   The last?   Anywhere in between?

Every ball has the same chance of being the first ball drawn, and $r$ of the $r+g$ balls are red.

Every ball has the same chance of being the second ball drawn, and $r$ of the $r+g$ balls are red.

$\vdots$

Every ball has the same chance of being the last ball drawn, and $r$ of the $r+g$ balls are red.

The counterinuition is that the colour of the first ball drawn influences the probability for the second ball, and this is true but only when you have knoweldge of what the first may be.   Without that conditional knowledge, each ball that was in the jar has the same chance to be the second ball drawn.

Solution 2:

Draw both balls with your eyes shut. Then look at the second ball before looking at the first ball.