I used about 400-500 tickets and had 3 dolls. It is likely that the doll falls on average every 100, but the first one appeared only after 150 and another after 50 - so it is random.
I've done the boss many times, I think even within 50 or more, and I didn't lose anything more than violet gem.
Now the event is more difficult, after today's patch so I think that the dolls will be worth more, but I think that every player has a chance at least one.