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In probability theory, a sample space is the set of all possible outcomes of a probabilistic experiment. It is often denoted by the symbol \( \Omega \). Whenever we define an experiment or a random process, determining the sample space is the crucial first step.

Consider rolling a six-sided die. The sample space \( \Omega \) in this case consists of the numbers that can appear face-up: \( \{1, 2, 3, 4, 5, 6\} \). Here, each outcome represents a single event that could occur when you roll the die.

Understanding the sample space helps us set the stage for calculating probabilities. It allows us to know every conceivable situation that might occur. In our exercise, \( \Omega \) represents the overarching set where every possible outcome resides. The subset \( A \) of this sample space includes outcomes we are particularly interested in analyzing. It's like focusing on a slice of the larger pie that is \( \Omega \).

This initial comprehension of the sample space and subsets like \( A \) enables deeper exploration into event probabilities and density functions later on.

Event probability refers to the chance of an event occurring within a sample space. An event is any subset of the sample space, and its probability is a measure of how likely this event is to happen.

Let's say we flip a coin. The sample space \( \Omega \) for this experiment is \( \{ \text{Heads}, \text{Tails} \} \). If we want to determine the probability of getting heads, we calculate the event probability \( P(A) \), where \( A \) is the event of flipping heads.

If each outcome is equally likely, the probability is given by dividing the number of favorable outcomes by the total number of possible outcomes. In the coin example, this would be \( \frac{1}{2} \), because there's one favorable outcome (heads) out of two possible outcomes (heads or tails).

In the context of our exercise, when we work with specific subsets \( B \) and \( C \) of \( A \), we analyze the probability of events occurring within these subsets. This forms the basis for comparing probabilities within a focused subset, allowing us to further understand and calculate \( P_1(B) \) and \( P_1(C) \), recalibrating our focus to the desired section of \( \Omega \).

For a function to be a valid probability density function, two crucial conditions must be met: non-negativity and normalization. Let's break these down.

• Non-negativity: This means that the probability assigned to any outcome must be zero or positive. It’s inherently illogical to have a negative probability because we can't have less than zero chance of something occurring. In our exercise, the function \( p_1(\omega) = \frac{p(\omega)}{P(A)} \) remains non-negative because as long as \( p(\omega) \) is a valid probability density function, \( p(\omega) \geq 0 \), and with \( P(A) > 0 \), it ensures \( p_1(\omega) \geq 0 \).


• Normalization: A probability density function has to sum up, or integrate, to one over its entire sample space. This requirement ensures that the total probability of some outcome occurring from the entirety of \( \Omega \) is always 100%. For \( p_1 \) in our problem, the normalizing step involves showing that \( \int_A p_1(\omega) \, d\omega = 1 \). We achieve this by verifying that dividing by \( P(A) \) correctly scales the probabilities so that they sum to one within subset \( A \).

These conditions ensure that we can interpret \( p_1 \) as a probability density function on its own, allowing for meaningful predictions and analyses to be made on the subset \( A \) within \( \Omega \). Understanding these principles is key to manipulating probability functions correctly and effectively.