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When exploring probability with two dice, we begin by understanding the sample space. Each die has six faces, which means when you roll one die, there are six possible outcomes. However, with two dice (like a red and a green one), we consider the outcomes of both dice together. Think of each die roll as an independent event, and then we pair up these two independent events to see all the possible results when both dice are thrown simultaneously.

Because each die has six possible outcomes, when two dice are rolled, there are a total of 36 possible outcomes. If we want to visualize it, we can imagine a grid with 6 rows and 6 columns, where each cell represents a different outcome with one die on one axis, and the other die on the other axis. Each outcome is written as an ordered pair (RedDie, GreenDie), representing the face-up numbers after the roll. For instance, the pair (1, 3) would denote the red die showing a 1 and the green die showing a 3 after the roll. Understanding this sample space is fundamental to computing probabilities for any events involving the roll of two dice.

To determine the likelihood of any specific event when rolling dice, you need to calculate the probability of that event. Probability is a measure of how likely it is for an event to occur and is given as a fraction or a percentage. The basic formula to calculate the probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

To illustrate, consider event A from the exercise: exactly one die is a 1. There are 10 outcomes where event A occurs, such as (1,2), (1,3), etc. Since the total number of possible outcomes in the sample space is 36, the probability of event A is therefore calculated as \(P(A) = \frac{10}{36}\). Similarly, we calculate the probability for event B, where the sum of the dice is even. Out of the 36 possible outcomes, 18 lead to an even sum, so \(P(B) = \frac{18}{36}\).

In probability, the intersection of events refers to the occurrence of two or more events at the same time. To calculate the probability of the intersection of two events A and B, denoted as \(P(A \cap B)\), we find the outcomes that are common to both A and B.

In our exercise, the intersection of event A (exactly one die is a 1) and event B (the sum is even) requires outcomes where one die is 1, and the sum is even. The outcomes that meet both these conditions are: (1,3), (1,5), (3,1), and (5,1). There are 4 such outcomes, so the probability of their intersection is \(P(A \cap B) = \frac{4}{36}\). Test for independence of two events A and B would require that \(P(A \cap B) = P(A) \times P(B)\). However, if \(P(A \cap B) \eq P(A) \times P(B)\), then events A and B are not independent. This final step is crucial in determining the relationship between two events within their sample space.