Stochastic process of ball into bins - "the most collided ball"

Question

Suppose I have balls in a bag and bins, for some integer >2. I draw a ball from the bag and throw it to one of the bins at random. If the bin is empty then the ball stays in the bin. However, if the bin is occupied by another ball then I take both of the balls out of the bin and put them back in the bag. Then I draw another ball from the bag and follow the same procedure. I keep doing this until the bag is empty (note that in this case all bins are either empty or occupied by one ball). What would be the probability that the ball that collided the most, will collide k times? I get quite confused when trying to solve this problem as the process itself is a random variable (perhaps it can help - @yuval-peres answered to me that the expected time until the process is finished is $n\frac{a}{a-2}$