First Draw
#8# blue balls and #5# black balls
The number of ways of getting #3# balls from #13# balls is
#=((13),(3))=(13!)/((13-3)!(3!))=(13*12*11)/(1*2*3)=286#
The number of ways of getting #3# black balls from #5# black balls is
#=((5),(3))=(5!)/((5-3)!(3!))=(5*4)/(2*1)=10#
The probability of getting #3# black balls is
#=10/286#
Second Draw
#8# blue balls and #2# black balls
The number of ways of getting #3# balls from #10# balls is
#=((10),(3))=(10!)/((10-3)!(3!))=(10*9*8)/(1*2*3)=120#
The number of ways of getting #3# blue balls from #8# black balls is
#=((8),(3))=(8!)/((8-3)!(3!))=(8*7*6)/(3*2*1)=56#
The probability of getting #3# black balls is
#=56/120#
Therefore,
The probability of getting #3# black balls and #3# blue balls is
#=10/286*56/120=7/429#
