1. Two dice are thrown. Find the probability that the sum of the points on the faces that come up is even, and that a six appears on at least one of the dice.
The problem is solved. On the emerged face of the "first" die, one point, two points, ..., six points can appear. Similar six elementary outcomes are possible when throwing the "second" die. Each outcome of throwing the "first" die can combine with each outcome of throwing the "second". Thus, the total number of possible elementary outcomes of the experiment is $6 \times 6 = 36$. These outcomes form a complete group and, due to the symmetry of the dice, are equally likely.

The favorable outcomes for the event of interest (at least one six appears on one of the faces, and the sum of the points is even) are the following five outcomes (the first number is the number of points that fell on the "first" die, the second number is the number of points that fell on the "second" die; the sum of the points is then calculated):

1) 6, 2; 6 + 2 = 8,
2) 6, 4; 6 + 4 = 10,
3) 6, 6; 6 + 6 = 12,
4) 2, 6; 2 + 6 = 8,
5) 4, 6; 4 + 6 = 10.

The desired probability is the ratio of the number of outcomes favorable to the event to the number of all possible elementary outcomes: $P = 5 / 36$.