manayay
Sep 23, 2013, 09:38 PM
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.
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.