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In a "double-six" domino set, understanding the combinations is crucial. A domino tile typically has two ends, each marked with a number of dots ranging from 0 to 6. These cards, or tiles, are a combination of every possible pair of numbers. The combinations range from (0,0) to (6,6).
Each of these pairs is unique in terms of the combination of numbers. To calculate them, it's essential to track down each unique pairing of numbers:

• (0,0), (0,1), …, (0,6)
• (1,1), (1,2), …, (1,6)
• (2,2), (2,3), …, (2,6)
• And so forth, up to (6,6)

This setup ensures that there are 28 unique tiles in a complete set, covering all combinations where the first number is less than or equal to the second number, ensuring no repeats occur. This concept is fundamental as it determines the number of total possibilities when calculating probabilities.

The total number of dots on a domino tile is simply the sum of the dots on both sides. For probability exercises, it's critical to understand how many dominos fit specific criteria related to their total dot count.
For instance, when we need to know how many dominos have a total of three dots or less, we look at these combinations:

• (0,0) with 0 dots
• (0,1) and (1,0), both with 1 dot
• (0,2), (2,0), and (1,1), each with 2 dots
• (1,2) and (2,1), each with 3 dots

Calculating these gives us a total of 7 dominos. Knowing how to identify these sums helps in determining which domino sets meet particular conditions needed for solving probability questions, such as whether a tile falls under a certain threshold of total dots.

To grasp probability in domino games, we must clearly define what constitutes a favorable outcome. This refers to the specific conditions that are met based on the problem's requirements. For example, when calculating the probability of a domino having a total dot count of three or less, our favorable outcomes are the 7 dominos identified by that criterion.
Calculating the probability requires dividing the number of favorable outcomes by the total number of possible outcomes (in this case, dominos). We have:

• Favorable outcomes: 7 dominos with ≤ 3 dots
• Total outcomes: 28

Therefore, the probability is calculated as: \[\frac{7}{28} = \frac{1}{4}\] This approach can be repeated for a variety of conditions, each defined by the specific requirements. Recognizing and calculating favorable outcomes are essential steps in more thoroughly understanding probabilistic concepts in dominos.