# I am indebted to Amin Sadeghi for this exercise. You must sort four balls into two buckets. There…

I am
indebted to Amin Sadeghi for this exercise. You must sort four balls into two
buckets. There are two white, one red and one green ball.

(a)
For each ball, you choose a bucket independently and at random, with
probability 1/2 . Show that the probability each bucket has a colored ball in
it is 1 2 .

(b)
You now choose to sort these balls in such a way that each bucket has two balls
in it. You can do so by generating a permutation of the balls uniformly and at
random, then placing the first two balls in the first bucket and the second two
balls in the second bucket. Show
»

I am
indebted to Amin Sadeghi for this exercise. You must sort four balls into two
buckets. There are two white, one red and one green ball.

(a)
For each ball, you choose a bucket independently and at random, with
probability 1/2 . Show that the probability each bucket has a colored ball in
it is 1 2 .

(b)
You now choose to sort these balls in such a way that each bucket has two balls
in it. You can do so by generating a permutation of the balls uniformly and at
random, then placing the first two balls in the first bucket and the second two
balls in the second bucket. Show that there are 16 permutations where there is
one colored ball in each bucket.

(c)
Use the results of the previous step to show that, using the sorting procedure
of that step, the probability of having a colored ball in each bucket is 2 /3 .

(d)
Why do the two sorting procedures give such different outcomes?

»

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