probability - If I reroll any 6s what are my chances of an EVEN result on a fair d6?

I have a normal, fair six sided die. I roll the die, but if I roll a 6 I keep rerolling the die until I get a result that is not a 6.

What are my chances of getting an even final result? Is it 2/5th? Or does it approach 2/5th?

I'm fairly certain that (ignoring the time it takes to roll the die) this scenario is the same as if I had a fair five sided die, but I don't know how to prove it.

Answer

It is indeed $\dfrac25$.

Suppose it is $p$. Then you can roll:

• two or four, with probability $\dfrac26$, and finally even


• one, three or five, with probability $\dfrac36$, and finally odd


• six, with probability $\dfrac16$, and then a further probability $p$ of being finally even

So the probability of being finally even is

$$p=\dfrac26+ \dfrac16p$$ which solves to $$p=\dfrac25$$