Additionally, when you throw the ball, it passes through a random accelerating mechanism that can make the ball pass in the box one, two or three times, with probabilities of 30%, 50% and 20%; each time thee ball re-enters the box, all the sticks are up, independently of the previous time

What is the probability of, in one shot, the maximum number of sticks being thrown down is 6? And 2?

And what is the probability of, in one shot, the minimum number of sticks being thrown down is 2?