Leadfoot has a 6/12 probability of winning
Farmerman has a 5/12 probability of winning
A draw has a probability of 1/12
You posed two questions and gave one solution.
Incorrect, the three probabilities are displayed.
That solution assigns a non-zero probability to an invalid ending with respect to the first question.
Your answer of 6/11 is incorrect the correct answer is 6/12.
Your recursive calculation does not account for redrawing when they both draw triangles.
Incorrect, the probability of redrawing is 1/12.
Your "somewhat more elegant" method is exactly what I did, but you, again, didn't treat the draw correctly for the first question.
Incorrect, the probabilities are as displayed, you forgot to account for the draw, if your first answer was correct it would have a numerator of 12.