There is a very classical understanding of probability where outcomes are simply counted. In that case, you might think the answer is 6/8 because six of the eight outcomes involve flipping T on the coin.
However, it seems unlikely that the probability of flipping a coin and getting tails is really 3/4. Assuming the coin is "fair," the probability of getting tails is 1/2 (and likewise, heads is also 1/2, since "fair" means the probabilities are equal).
You see, you cannot simply count six out of eight outcomes unless all the outcomes are equally likely (they are not -- some have probability 1/4, some 1/12). Even though the coin is "fair" and has equally likely outcomes, just as the die does, the probabilities are different (1/2 for the coin, 1/6 for the die).
But the answer is simple, you roll the die exactly when you flip the coin and get tails on the first flip -- which happens with probability 1/2.
You can also compute the probabilities of each event, see attached. You'll notice, not only do you see 1/2 leading up to the first T flip (green), but the sum of probabilities of all possible outcomes that involve a die being rolled (blue) is also:
1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12 = 1/2
which agrees with the 1/2 we had from looking to the green node.