### Infinite Balls and Jugs

Posted:

**Thu Sep 17, 2009 6:14 pm UTC**Suppose I have infinitely many balls, numbered 1,2,.. and so on.

At 10 minutes to midnight, I put balls 1-10 in the jug, and remove a ball. At 5 minutes to midnight I put balls 11-20 in the jug and remove a ball. At 2.5 minutes to midnight, etc (halving the time between insertions ad infinitum, putting the next 10 balls numerically in the jug each time).

How many balls are in the jug at midnight if:

a) I always remove the lowest numbered ball

b) I always remove the highest numbered ball

c) I remove balls uniformly at random.

Solution thread.

Hints:

At 10 minutes to midnight, I put balls 1-10 in the jug, and remove a ball. At 5 minutes to midnight I put balls 11-20 in the jug and remove a ball. At 2.5 minutes to midnight, etc (halving the time between insertions ad infinitum, putting the next 10 balls numerically in the jug each time).

How many balls are in the jug at midnight if:

a) I always remove the lowest numbered ball

b) I always remove the highest numbered ball

c) I remove balls uniformly at random.

Solution thread.

Hints:

**Spoiler:**

**Spoiler:**

**Spoiler:**