I read that you are looking for easier method , so the suggestion is to use generating functions .We know that the die (standard) can have only the values of $1,2,3,4,5,6$. Then , the generating function for die is :$$x+x^2+x^3+x^4+x^5+x^6 = \frac{x-x^7}{1-x}$$
Now , we have $12$ dice , so if we find the product of these $12$ dice such that $$\bigg(\frac{x-x^7}{1-x}\bigg)^{12}$$ the sum of coefficient ranging from $x^{12}$ to $x^{29}$ will give us the number of possible occasions such that the summation over the dice less than $30$.Because of summing the values in the link is cumbersome , i did not write the exact solution , but if you write a computer program that can calulcate sum the coefficent of the values whose exponents from $x^{12}$ to $x^{29}$ and sum the coefficent of the values whose exponents from $x^{12}$ to $x^{72}$ , then when you divide them it will give you the probability