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In probability theory, mutually exclusive events are events that cannot happen simultaneously. This characteristic means that the occurrence of one event rules out the occurrence of the other. For example, when you flip a coin, the results 'heads' and 'tails' are mutually exclusive because you can only get one result per flip.

For two events, say \(A_1\) and \(A_2\), being mutually exclusive, their intersection is empty, denoted as \(A_1 \cap A_2 = \emptyset\). This simply means there are no outcomes that are common to both events. In essence, when one event occurs, it is impossible for the other event to occur. Understanding this concept is crucial while working with probability equations and helps in simplifying calculations.

• If events are mutually exclusive, \(P(A_1 \cap A_2) = 0\).
• The probability of either \(A_1\) or \(A_2\) happening is the sum of their individual probabilities: \(P(A_1 \cup A_2) = P(A_1) + P(A_2)\).

The sample space, denoted as \(S\) in probability, is the set of all possible outcomes of a particular experiment or random trial. For instance, when rolling a six-sided die, the sample space is composed of the numbers \(\{1, 2, 3, 4, 5, 6\}\).

Understanding the sample space is fundamental, as it forms the foundation for calculating probabilities. The probability of the entire sample space, \(P(S)\), is always equal to 1. This is because the sample space represents every possible outcome, ensuring at least one outcome will happen in every trial.

• In the problem presented, the union \(A_1 \cup A_2\) is equal to the sample space, \(S\), indicating at least one of these events must occur in every possible outcome.
• The equation \(P(A_1 \cup A_2) = 1\) supports this understanding because it states that either \(A_1\) or \(A_2\) will happen.

Probability equations are mathematical formulations that help calculate the likelihood of events. They are quite useful in solving problems involving mutually exclusive events and sample spaces.

In the given problem, two main probability equations are identified:

• The first equation is derived from the characteristic of the sample space: \(p_1 + p_2 = 1\). This equation denotes that the sum of the probabilities of events \(A_1\) and \(A_2\) must equal the probability of the sample space.
• The second equation, \(3p_1 - p_2 = \frac{1}{2}\), adds a condition that further defines the relationship between \(p_1\) and \(p_2\).

To solve these types of equations, basic algebraic methods like substitution or elimination are used to find the unknown probabilities. By solving these specific equations together, you can determine the individual probabilities of \(A_1\) and \(A_2\), which are essential for understanding the problem's overall context and its solutions.