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When discussing probability, the sample space is a fundamental concept. It represents the set of all possible outcomes for a given experiment or event. In the context of rise times for batches in a reactor, the sample space consists of all possible positive real numbers, since rise times can vary infinitely.
For two batches, this sample space extends to a two-dimensional realm, with each axis representing the rise time of one batch. This results in a plane where all points have positive coordinates, specifically describing the first quadrant of a Cartesian plane. This concept is crucial as it sets the boundary for analyzing events and their probabilities within this space. Understanding the sample space helps in visualizing how different conditions isolate parts of this space for specific events.

A two-dimensional plot provides a visual representation of the sample space for two variables. For our case, these variables are the rise times of batch 1 and batch 2. The x-axis illustrates the rise time for batch 1, while the y-axis illustrates that for batch 2.
When you plot sample space on a two-dimensional graph, each point's coordinates correspond to a specific pair of rise times from the batches. This kind of visualization helps to better understand complex relationships between multiple events and to easily mark conditions or changes in those relationships. A two-dimensional plot is particularly useful in probability because it can show continuous outcomes, such as real numbers, as well as discrete ones.

Complementary events in probability are pairs of events whose probabilities sum up to 1. If one occurs, the other does not, and vice versa. For instance, if event B is the rise time of batch 2 being greater than 52.5 minutes, its complement, denoted as B', is the rise time being less than or equal to 52.5 minutes.
On a two-dimensional plot, complementary events are visually depicted as non-overlapping regions that together cover the entire sample space related to the event in question. Understanding complementary events is vital because it allows for calculating probabilities by subtracting from 1, which can simplify complex probability problems.

The intersection and union of events help us understand combined probabilities and relationships between different events.

• Intersection (\( A \cap B \): This refers to the event where both A and B occur simultaneously. On a graph, this is depicted as the overlapping area between the two respective regions. For example, in our exercise, this would be the region where the rise time of batch 1 is less than 72.5 minutes and batch 2 is greater than 52.5 minutes.
• Union (\( A \cup B \): The union signifies at least one of the events occurs. On the plot, this covers all areas encapsulated by either A or B, or by both. Visually, it’s a blend of individual regions, showing situations where either or both conditions of A and B are fulfilled.

Intersection and union provide powerful insights into how various conditions overlap and contribute to the overall probability insights.