I have a very simple question about the act of being able to re-rolling a single die if it fails to produce the correct result and how that may or may not change the probability of the outcome. I've searched the infinite number of internet pages on dice odds and haven't found a clear example.
If I roll a single die then the odds of rolling a 5 or 6 is 2/6 or 33.33%...right? I hope I got this part correct so far! Now, if I'm allowed to re-roll the die a single time if I don't roll a 5 or 6 then how does a single re-roll affect the probability? Basic instincts tell me that the die still has a 33.33% chance on the second roll, but since a second roll is dependent on the outcome of the first would that change the probability?
Doing some basic examples reveals a probability, but I don't know if its a fare example to the above questions...If I roll 36 dice, then in a perfect world of probability 12 dice would have a 5 or 6. Now, 24 of the dice did not, so I'd be allowed a single re-roll of those dice, which would result in 8 more 5 or 6s. Once all failed dice have been allowed a re-roll I'm left with 20 dice with 5 or 6s from the original 36. That's 55% of the dice...
Back to a single die. Does the above sample suggest that the simple act of re-rolling once on a failed roll would increase a dies chances by 22% (from 33% to 55%!? That seems suspiciously high. If this is true (or not) can some one please explain the math involved for calculating the probability?
Thanks so much for any clarification!