Three Buckets Have Marbles

You can justify if you write the probability equations.

Three buckets have marbles. Pour the remaining water from 5l bucket to 8l bucket. Water in 8l bucket is now 2l. If it does not tip to a side if it balances we know the unused group of 3 contains the heavy marble. 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.

That means that the probability of picking bucket 3 is 1 3 or 33. X y z are integers. 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 have the probability p1 for one bucket to take in the first attempt a white marble the same for a probability p2.

Prove that it is possible to produce an empty bucket with a series of such operations. Given 3 buckets containing x y z litres of water. I m going with the latter since the topic of your question suggests it. 1 1 when we are allowed to throw water out of bucket.

Water in 5l bucket is now 2l. You are allowed to double the number of marbles in a bucket by borrowing from one of the other two buckets. So all moves have to of the form. 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.

Fill the 5l bucket full. Pour the water from 5l bucket to 3l bucket. Then we say we get another 18 marbles and distribute those additional marbles evenly into the same three buckets. The probability of any given marble being white can be.

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. Assume buckets have large size. If the scale tips to a side we know that group of 3 has the heavy marble. Since we now have 3 marbles 1 of which is the heavy one put 1 in each side 1 left 1 right 1 unused.

Repeat step 1 to 3 again and we will have 4l of water in 8l bucket. And we represent that procedure by telling students that 12 18 div 3 12 div 3 18 div 3.