Question

Suppose that 3 balls will be randomly put into 3 buckets, with each ball being equally likely to be put into each of the buckets, independently of which buckets are chosen for any of the other balls. (E.g., the first ball is equally likely to be put into any of the 3 buckets, and then regardless of which bucket the first ball is placed in, the second ball is equally likely to be put into any of the 3 buckets.) What is the probability that all of the balls will be put into the same bucket

Answer 1

`(1)/(27)`

Explanation:

Since there are a total of three buckets and only one can be chosen at a time, this would mean that the probability of a ball being placed in a bucket is 1/3. Since each ball has the same probability of being placed into any bucket regardless of the where the previous ball landed, it means that each ball has the same 1/3 probability of a bucket. In order to find the probability that all three land in the same bucket, we need to multiply this probability together for each one of the balls like so...

`(1)/(3) * (1)/(3) * (1)/(3) = (1)/(27)`

Finally, we see that the probability of all three balls landing in the same bucket is
`(1)/(27)`