Three Buckets Have Marbles

Pour the water from 5l bucket to 3l bucket.

Three buckets have marbles. If it does not tip to a side if it balances we know the unused group of 3 contains the heavy marble. You have the probability p1 for one bucket to take in the first attempt a white marble the same for a probability p2. Keep the heavy group of 3 marbles and discard the rest. Water in 8l bucket is now 2l.

Then we say we get another 18 marbles and distribute those additional marbles evenly into the same three buckets. Given 3 buckets containing x y z litres of water. Three buckets have marbles. So all moves have to of the form.

Prove that it is possible to produce an empty bucket with a series of such operations. Water in 5l bucket is now 2l. And we represent that procedure by telling students that 12 18 div 3 12 div 3 18 div 3. Repeat step 1 to 3 again and we will have 4l of water in 8l bucket.

Fill the 5l bucket full. We tell students that we have 12 marbles that we want to put in three buckets with an equal number of marbles in each bucket. You can justify if you write the probability equations. Since we now have 3 marbles 1 of which is the heavy one put 1 in each side 1 left 1 right 1 unused.

You are allowed to double the number of marbles in a bucket by borrowing from one of the other two buckets. Assume buckets have large size. We assume marbles are equally to be taken and the total amount is the same in the two buckets where the total is 2m where m is the number of total white marbles and the same quantity for black marbles and m is too the. Each bucket has k l m marbles respectively k l m n i will pick one marble from each of the three buckets with probability 1 k 1 l 1 m respectively and i ll put them into a another bucket.

The probability of any given marble being white can be. That means that the probability of picking bucket 3 is 1 3 or 33. If the scale tips to a side we know that group of 3 has the heavy marble. X y z are integers.

What i am looking for is the probability that i should assign to each of the three marbles so that i get to choose one uniformly at random 1 n. 1 1 when we are allowed to throw water out of bucket.