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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