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Rolling dice is a classic scenario often used in probability exercises. When you roll a single six-sided die, there are six possible faces that can land up: 1, 2, 3, 4, 5, or 6. Each face has an equal chance of 1/6 of showing up.

If you roll more than one die, like in this exercise, the number of possible outcomes increases significantly. For instance, rolling two dice yields 36 possible combinations, as each die operates independently. The calculation follows this principle: for each face of one die, all six faces of the second die could also appear. Therefore, you multiply: 6 x 6 = 36.

Likewise, rolling three dice leads to 216 possible outcomes (6 x 6 x 6). Understanding these basics helps you calculate the probability of specific events when dealing with dice.

Triples are an interesting outcome when rolling multiple dice. In the context of dice, a triple occurs when all three dice result in the same number. This means, if you roll three dice, potential triples are combinations like (1,1,1), (2,2,2), and so on up to (6,6,6).

Given there are six faces on a die, there are six possible triplets. Hence, the number of favorable outcomes for rolling triples is simply six.

In probability terms, it's crucial to identify how these favored events relate to the entire space of outcomes, which is 216 for three dice, as calculated previously. So, the probability of rolling triples is: \[ P(\text{triples}) = \frac{6}{216} = \frac{1}{36} \]

This means that there is a 1 in 36 chance of rolling the same number on all three dice.

Calculating the probability of rolling a particular sum with three dice is more complex than counting triples. For example, to achieve a sum of 5 with three dice, you must examine various combinations of numbers that add up to exactly 5.

Possible sets include:

• \((1, 1, 3)\): This can be arranged in 3! = 6 ways.
• \((1, 2, 2)\): This can be arranged using \(\frac{3!}{2!} = 3\) ways.

Adding these, there are 9 ways to make a sum of 5.

Given the total number of outcomes is 216, as established when determining total possible rolls, the probability for rolling a sum of 5 is:
\[ P(\text{sum of 5}) = \frac{9}{216} = \frac{1}{24} \]

This shows that you have a 1 in 24 chance of rolling a sum of 5 with three dice.

In probability, the idea of outcomes deals with understanding the various possible events that can occur during an experiment, like rolling dice. Each individual result, or combination of dice faces, is referred to as an outcome. Probability is all about determining how frequently a specific outcome results among all the possible outcomes.

When you calculate the probability of an event, you divide the number of favorable outcomes (those matching the criteria of your event) by the total number of outcomes. Using dice, where each die has 6 independent faces, outcomes become a multiplication of possibilities. For example, with three dice, it results in \(6 \times 6 \times 6 = 216\) possible outcomes.

Understanding this framework helps in sorting probability into clear formulations, making it easier to calculate like how we did for triples and the sum of numbers. This structured approach allows you to organize any problem into systematic steps for finding probabilities.