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Dice probability involves understanding how likely different outcomes are when rolling dice. When you roll two six-sided dice, each die has possibilities numbered 1 through 6. This means, every roll of the dice is an independent event, and each number has an equal chance of appearing. Therefore, the probability for each possible result depends only on the number of ways we can achieve that result, divided by the total number of possibilities.

In this exercise, we are interested in the specific event where one die shows a 6, and the other shows a number less than 3. Understanding dice probability helps us determine the frequency of these specific outcomes. Exploring dice probability gives us insight into how random events behave and can be applied in games and real-life decision-making.

The concept of a sample space is crucial when dealing with probability, as it represents all possible outcomes of an experiment.

For this exercise, since we are rolling two dice, the sample space refers to all possible pairs of numbers we might see. Each die has 6 faces, so there are 6 possibilities for the first die and 6 for the second, leading to a total of 36 possible outcomes. These outcomes can be written as ordered pairs (x, y), where x and y are the numbers on each die. For example, (1,1), (1,2), ..., (6,6).

Having a clear understanding of the sample space ensures that we don't overlook any possible outcomes, allowing us to calculate probabilities accurately. Knowing how to enumerate the sample space is key when solving probability problems involving dice.

Desired outcomes are the specific results we want to observe in a probability experiment. These are the cases that we are interested in when calculating the probability of a particular event.

For this setup with two dice, we are looking for scenarios where one die shows a 6, and the other shows a number less than 3. There are only four such possible outcomes: (6, 1), (6, 2), (1, 6), and (2, 6). These outcomes match our condition perfectly. It’s essential to correctly identify these desired outcomes as they directly influence the probability calculation.

Highlighting the importance of desired outcomes is necessary because making any mistakes here can lead to incorrect probability calculations. They form the basis for any further probability computations.

Probability calculation involves determining the likelihood of a specific event occurring within the sample space. The probability of an event can be found using the formula:

• Probability = (Number of desired outcomes) / (Total number of possible outcomes)

Applying this formula, we first identified that there are 4 desired outcomes (as determined in the previous section) and 36 total possible outcomes (from the sample space). Thus, the initial probability is given by:

• \(P(E) = \frac{4}{36}\)

To simplify the probability, we need to divide the numerator and the denominator by their greatest common divisor (GCD), which is 4 in this case. This reduces the probability to:

• \(P(E) = \frac{1}{9}\)

This means, there is a 1 in 9 chance for one die to show a 6 and the other to show a number less than 3 when rolling a pair of dice. This straightforward method of calculating probabilities provides a foundational understanding that can be applied to a variety of situations involving random events.