I have 3 bags. Red, blue and green bag.

Red and Blue bags have 6 balls each.

Red’s balls are numbered 1 to 6

Blue’s balls are numbered 7 to 12

Green bag has 9 balls, numbered 13 to 21.

First thing I do is I take out one ball from each of the bags and the 3 balls are put in a small purple bag. All other balls are put all together into a bigger white bag.

Then 6 people pick 3 balls each - without seeing them so you cannot tell how many per color (you can end up with 3 red, or 2 blue and a green for example) - from the white bag.

Now, let’s think at ball 1, originally in the red bag.

I can say that the probability for it being in the purple bag is 16.66% (1/6).

But what is the probability for it to belong to person number 1?

It’s here that I am lost.

If I knew (but I don’t!) that this ball is

*for sure*in the white bag, and I only consider the probability to be picked by one person, then am I right to say that each ball has a 3/18th of chance (so still 16.66%) to be picked.

But because first of all I can't tell if ball 1 is in white bag and if that is not the case, then it can be picked by someone else, how can I tell each ball’s probability to belong to each person?

I hope I made sense. I would love to see a formula for it, cause I want to see how this % changes if I change the variables

variables are for example,

I know that the ball is for sure in the white bag or

I know that the first two people did not pick it or

Person number 4 has already picked two balls that are not number 1 (during the game, balls are slowly revealed)

I think it's a quite easy problem for you experts, possibly so a very big thank you in advance to whoever can help me!