Conditional Probability, Independence & Bayes · Superchallenge problem

A bag holds four dice: a $4$-sided, a $6$-sided, an $8$-sided and a $12$-sided die, each fair and numbered from $1$ up to its number of sides. One die is taken from the bag at random and rolled twice, showing $3$ both times. If the same die is rolled a third time, what is the probability that it shows $3$ again? Express your answer as a common fraction.