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.

  • $\begingroup$ 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. $\endgroup$ – 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 $$p=\dfrac25$$


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