Loaded dice and unfair polynomials

If you throw a pair of fair dice, you are six times more likely to throw a total of 7 than a total of 12. This is because the only way to throw a 12 is for each of the dice to show a 6, as in the picture, whereas there are six ways to throw a 7: 1+6, 2+5, 3+4, 4+3, 5+2, and 6+1. The chart below shows how many ways there are to throw each possible total;…