the problem of the 3 balls

[thoandros]

Hello everyone. I have a problem, or at least, I am stuck to find a formula that would work and apply all the time to find the probability for each ball to be possessed by a certain person. Here the explanation:


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!

 

[Dr.Peterson]

In order for ball 1 to have been picked by person 1, it must first be in the white bag (with what probability?), and then it must be picked by person 1 (with probability 3/18 = 1/6, as you said). What is the resulting probability?

What balls others get does not affect this probability. (But if you knew what the others picked, then you would be asking about a conditional probability, which of course would be different.) Ultimately, you just have 7 sets of 3 balls, and are asking for the probability that a given ball is in a given set, which is easy.

 

[thoandros]

In order for ball 1 to have been picked by person 1, it must first be in the white bag (with what probability?), and then it must be picked by person 1 (with probability 3/18 = 1/6, as you said). What is the resulting probability?

The probability for ball 1 to be in white bag is 5/6. Because into the purple bag there can only be one ball from the red, blue and green bags.

Ultimately, you just have 7 sets of 3 balls, and are asking for the probability that a given ball is in a given set, which is easy.

Yes, well, I think that is the starting point.

However the ultimate goal is to establish a formula that considers the variables (so yes, I think that is a conditional probability case) and keeps updating the probability for each ball.

For example, if I know for sure that ball 3 (from red bag) is now in the purple bag, then I can be 100% sure that ball 1 will be picked by one of the 6 people. Let's call them A to F, ok?
Now, still for example, if I know all the 3 balls persons A,B,C & F have, and based on the fact above, I can tell for sure that ball 1 is 50% in possess of person D or E.
Now let say that I know already 2 of the 3 balls of person D, but I know none of those held by E.
How would this impact the % for Ball 1 to be held by D, considering that among all the balls to be "assigned", D only has 1 "slot" while E has 3 slots.

If I knew 2 out of 3 balls for both D or E, I can tell for sure that Ball 1 is 50%held by D and 50% by E. But not knowing that for E, how would my % change? I mean, I know that can be D or E only, but I don't think it's a 50:50.

Am I making sense?