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Understanding probability theory is essential when dealing with uncertain events like rolling dice. Probability is the branch of mathematics that studies the likelihood of events occurring. It quantifies uncertainty and allows us to make predictions about the results of various random processes.

In the context of dice rolling, probability helps us answer questions like: What is the chance of rolling a double six? The calculation of this takes into account all possible outcomes and the specific outcome we are interested in. The theory dictates that probability values are between 0 and 1, where 0 means an event is impossible and 1 means it is certain. This context falls perfectly with our exercise, where we computed the likelihood of rolling a pair of 1s.

The probability of an event is calculated using the formula: \[ Probability = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} \] This principle is fundamental when it comes to providing a structured approach to finding probabilities and is directly used in our step-by-step solution for the exercise in question.

Dice probability refers to the chances of rolling a certain number or combination of numbers with a dice or a pair of dice. A standard die has six faces, numbered 1 to 6. When a single die is rolled, each outcome (1 through 6) has an equal chance of appearing; this is known as a uniform probability distribution.

However, when dealing with two dice, as in our exercise, it's important to understand that while there are 6 outcomes for one die, the total outcomes for two dice are 6 times 6, or 36, because each die is independent of the other. This independence is why we can multiply the possibilities of the first die with that of the second to get the total outcome possibilities as shown in Step 1 of the solution.

To further clarify the concept, each die is independent, leading to the fundamental counting principle, which explains the 36 different outcomes. So, while rolling a single 1 has a probability of 1/6, rolling a pair of 1s, as needed for our exercise, the probability is much less, specifically 1/36 as illustrated in the final solution.

The fundamental counting principle is a foundational rule in probability that allows us to count the number of ways an event can occur. It states that if there are 'n' ways for one event to occur, and 'm' ways for another event to occur, then there are n times m total ways for both events to occur together.

To apply this principle to our dice problem: Rolling one die presents us with 6 outcomes, and rolling a second die, independently, also presents 6 outcomes. The principle then tells us to multiply these two (6 x 6) to find the total number of possible outcomes, which is 36 for two dice.

Another way to visualize the fundamental counting principle is by creating a grid with all possible outcomes. If we made a grid for our two dice, one axis would represent the outcomes of the first die, and the other axis would represent the outcomes of the second die. Each cell of the grid represents a unique outcome of the pair of dice. It is this simple yet robust principle that plays a crucial role in the calculation of probabilities in various scenarios—including the dice roll probability we have explored in the exercise.