What color of ball do you want?

The fundamental act is that we draw 6 balls together. The number of outcomes for this act is $_{62}C_{6}=61441519$ .

The number of ways of drawing 6 red balls is $_8C_6=28$ , 6 yellow balls is $_{10}C_6=210$ , 6 blue balls is $_{13}C_6=1716$ , 6 green balls is $_{16}C_6=8008$ , 6 white balls is $_9C_6=84$ and 6 violet balls is $_6C_6=1$ . The number of ways of drawing 6 balls of the same colors is $28+210+1716+8008+84+1=10047$ .

Hence, the probability is

$p=\dfrac{10047}{61474519}\approx 0.0001634$