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.

That means that the probability of picking bucket 3 is 1 3 or 33. So all moves have to of the form. Water in 8l bucket is now 2l. 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. You are allowed to double the number of marbles in a bucket by borrowing from one of the other two buckets. You have the probability p1 for one bucket to take in the first attempt a white marble the same for a probability p2. 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.

The probability of any given marble being white can be. Water in 5l bucket is now 2l. 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. Given 3 buckets containing x y z litres of water.

Then we say we get another 18 marbles and distribute those additional marbles evenly into the same three buckets. 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. 1 1 when we are allowed to throw water out of bucket.

X y z are integers. 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. I m going with the latter since the topic of your question suggests it. Repeat step 1 to 3 again and we will have 4l of water in 8l bucket.

Keep the heavy group of 3 marbles and discard the rest. Pour the remaining water from 5l bucket to 8l bucket. Three buckets have marbles. 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.

If it does not tip to a side if it balances we know the unused group of 3 contains the heavy marble. Fill the 5l bucket full.