Three Buckets Have Marbles

And we represent that procedure by telling students that 12 18 div 3 12 div 3 18 div 3.

Three buckets have marbles.

Pour the remaining water from 5l bucket to 8l bucket. You are allowed to double the number of marbles in a bucket by borrowing from one of the other two buckets. Fill the 5l bucket full. 1 1 when we are allowed to throw water out of bucket.

You can move water from one bucket to another only if the bucket to which water is being transferred doubles the amount of water it has. 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. If the scale tips to a side we know that group of 3 has 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.

Assume buckets have large size. Given 3 buckets containing x y z litres of water. Pour the water from 5l bucket to 3l bucket. That means that the probability of picking bucket 3 is 1 3 or 33.

Keep the heavy group of 3 marbles and discard the rest. So all moves have to of the form. X y z are integers. Since we now have 3 marbles 1 of which is the heavy one put 1 in each side 1 left 1 right 1 unused.

If it does not tip to a side if it balances we know the unused group of 3 contains the heavy marble. Then we say we get another 18 marbles and distribute those additional marbles evenly into the same three buckets. 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. Water in 5l bucket is now 2l.

Repeat step 1 to 3 again and we will have 4l of water in 8l bucket. The probability of any given marble being white can be. You can justify if you write the probability equations. 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.

Three buckets have marbles. Prove that it is possible to produce an empty bucket with a series of such operations.