# 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.

• Your intuition is correct. In terms of the possible outcomes, rolling a six is the same as not rolling the die at all. This is a common way to generate a uniformly-distributed random integer on an interval that’s smaller than the one that your random number generator covers: discard any results that are out of range and try again. – amd Aug 31 '16 at 18:49

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 $$p=\dfrac25$$