A specially made pair of dice has only one-and two-spots on the faces. One of the dice

Question:

A specially made pair of dice has only one-and two-spots on the faces. One of the dice has three faces with a one-spot and three faces with a two-spot. The other die has two faces with a one-spot and four faces with a two-spot. One of the dice is selected at random and then rolled six times. If a two-spot shows up only once, what is the probability that it is the die with four two-spots?