How many even numbers will $99$ dice show if we roll them for eternity under a certain condition?

Question

Consider a six-sided die with numbers from $1$ to $6$. Imagine you have a jar with $99$ of such dice. You throw all dice on the floor randomly. You look at one of the dice on the floor at a time. For each die, you do the following:

• If it landed at an even number $(2,4,6)$, you turn the die so that it lands on the number $1$.

• If the die landed on an odd number $(1,3,5)$, you throw the die up in the air, so it can land on any number.

After you finish doing the above for all dice on the floor, you come back to the first die and repeat the entire process again. You keep on doing this until eternity (for a billion years, let’s say). If I come into the room after a billion years, how many dice on the floor will have even numbers up?

Answer 1

Each time you go through, you change every even die into an odd one, and you turn half of the odd dice into even ones. These two processes are in balance only if a third of your dice are even.

Answer 2

If $X_{n}$ denotes the number of dice landed on even numbers after $n$ rounds and $X_{0}:=0$ then:

$$\mathbb{E}X_{n}=\frac{1}{2}\left(99-\mathbb{E}X_{n-1}\right)$$

This equation tells us that: $$\lim_{n\rightarrow\infty}\mathbb{E}X_{n}=33$$