Dice Probability: Probability of getting a 6 on three dice if rolls of 1-2 are rerolled

Question

Question: As above, I'm currently dealing with a situation where three dice are thrown and a single six is needed for success, re-rolling 1s or 2s. I understand how this works in principle without the re-roll, getting 91/216 as the probability of a 6 on the three dice but don't get exactly how to factor in said re-roll because it introduces dependent probability. I'm also interested in finding the probability distribution for more than one 6 given the same conditions but that's secondary to understanding the first part.

Answer 1

If you only reroll once, then the probability for obtaining a six on a particular die is that of doing so on the first roll or needing to reroll and obtaining a six on that.

$$\tfrac 16+\tfrac 26\tfrac 16=\tfrac {2}{9}$$

Beyond that, the principle is the same.   The count of sixes rolled among the three dice follows a Binomial distribution.   The reroll process merely increases the probability for success; that is obtaining a six on any particular dice; since you may have a second chance.   The events for obtaining a six on separate dice remain independent and identically probable.