Let's take the case of picking first.

You can win with your second try or your third try...

The probability of winning on YOUR second pick meant that you picked red, your opponent picked red

and you picked red again.

The probability of winning on YOUR third pick meant that on the first four selections there were two red and two blues

AND then you picked the third red on the fifth overall selection.

and so on.....

These fractions reduce, but I wanted to show the pattern.

They reduce to

where n=1,2,..9, since your last pick is the 19th and we need that to be red and have conditions placed on the first 18 balls. So my answer seems to be