Even Rolls Until Six

Conditioning on "all rolls even" is not the same as rolling a die with faces $\{2,4,6\}$ — long sequences are much less likely to survive the condition.

Stop the process at the first roll in $\{1,3,5,6\}$. The condition says that stopping roll was a 6. Is the stopping time independent of which of these faces stopped it?

1. 1. Avoid the trap

   Restricting each roll to $\{2,4,6\}$ would give $3$, but that ignores that the event "no odd roll before the 6" is more likely for short sequences.

2. 2. Reframe the stopping time

   Let $T$ be the first roll showing $1,3,5$ or $6$. Then $T$ is geometric with success probability $4/6=2/3$, and the event is exactly "roll $T$ shows a 6".

3. 3. Use independence

   The face at time $T$ is uniform on $\{1,3,5,6\}$ independently of $T$, so conditioning on it being a 6 leaves the distribution of $T$ unchanged.

   $$\mathbb{E}[N\mid \text{all even}]=\mathbb{E}[T]=\frac{1}{2/3}=\frac{3}{2}$$

Answer: 3/2