I have 5 identical balls and I have three buckets which are
labelled A, B, C.
(a) In how many different ways can I put the five balls into the buckets.
(There are no constraints, e.g. a bucket can be empty.)
(b) In how many different ways can I put the five balls into the buckets if
each bucket must contain at least one ball?
Now suppose I perform the following experiment. For each ball I pick one of
the three buckets with equal probability and put the ball in that bucket. I
claim that the probability that each bucket has at least one ball is 50/81. One
might think this probability would be given by your answer to (b) divided
by your answer to (a), which should work out to 2/7.
(c) Explain why your answer to (b) divided by your answer to (a) is not the
correct probability.
(d) Derive my answer.