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Understanding probability calculation is essential in the realm of statistics and gambling, particularly when it comes to games involving dice, like the scenario of rolling a pair of dice three times. Probability quantifies the likelihood of a certain outcome occurring out of all possible outcomes. Simplified, it's the count of 'successful' outcomes divided by the total count of all possible outcomes.

In our dice-rolling scenario, to find the probability of a specific event, like rolling exactly one 7 in three tries, we have to consider the number of ways we can achieve this outcome versus the total number of outcomes possible in three rolls.

Following the given solution, calculating the probability involves recognizing that there are multiple sequences where exactly one 7 can occur: once in the first roll, the second, or the third. Each outcome has a calculated probability, derived from the combination of rolling a 7 and not rolling a 7. In this instance, the problem can be simplified by multiplying the probabilities of these individual events for each sequence, and then adding these probabilities together to get the total probability.

Combinatorics plays a pivotal role in probability, particularly when dealing with multiple events or choices. It's the field of mathematics focused on counting, both in a concrete and abstract sense, and especially useful when determining the number of possible outcomes in a probabilistic event.

When looking at dice, combinatorics helps us understand that there are 6 faces on a die and thus 36 (6 x 6) possible outcomes when two dice are rolled simultaneously. From these 36 outcomes, we determine that 6 outcomes sum up to 7. The probability for one event is then the favorable outcomes (rolling a 7) divided by all possible outcomes (all possible dice combinations).

The use of combinatorics in the solution shows us how to count the number of favorable sequences (YNN, NYN, and NNY) and takes into account that these sequences are equally likely due to the nature of independent dice rolls. By applying the principle of combinatorics, we were able to break down a more complex problem into manageable parts, greatly simplifying the probability calculation.

In the study of probability, understanding independent events is critical. Two or more events are independent if the outcome of one event does not influence the outcome of another. For dice rolls, each throw is independent of the others; the result of one roll doesn’t impact the result of the next roll.

This concept is crucial for our exercise because it allows us to multiply the probabilities of each individual sequence event to find the overall sequence probability. As demonstrated in the solution, the probability of not rolling a 7 after a 7 has been rolled remains unchanged (30/36), despite the previous event.

This independence simplifies calculations, as seen with the multiplication of the individual event probabilities to find the total sequence probabilities. In summary, the essence of independent events in probability is that each event occurs in its own bubble, uninfluenced by the others, making the mathematical analysis much more straightforward.