I do NOT have a 1 in 10 chance of getting any BG on every queue.
I didn't say it was an even chance. These probabilities would be based on 10,000 people queuing and measuring who gets into what. You, the individual, could still luck (or unluck) out and get the same BG every single time even if it's a supposed 3% chance.
My point was that hitting random will never be as random as you so desire. You don't like it because you wanted variety AND the bonus honour that comes with clicking random.