The number of combinations to take 10 balls out of 20 is 20!/(10!)^2=184756.
The number of combinations to take 5 balls of each color is (10!/(5!)^2)^2=63504.
This means in 184756 games you lose 63504 games and win 121252 games in average, totaling a loss of $5756. In other words, you lose about $0.03116 per game in average.
Therefore, the gamble is not fair.
Today's brain teaser courtesy of Braingle.com.