Problem 33 - Applied Statistics and Probability for Engineers

Question

A reactor's rise time is measured in minutes (and fractions of minutes). Let the sample space for the rise time of each batch be positive, real numbers. Consider the rise times of two batches. Let \(A\) denote the event that the rise time of batch 1 is less than 72.5 minutes, and let \(B\) denote the event that the rise time of batch 2 is greater than 52.5 minutes. Describe the sample space for the rise time of two batches graphically and show each of the following events on a twodimensional plot: (a) \(A\) (b) \(B^{\prime}\) (c) \(A \cap B\) (d) \(A \cup B\)

Step-by-Step Solution

Answer

See graph for events: \(A\) is left of \(x=72.5\); \(B'\) is below \(y=52.5\); \(A \cap B\) is left of \(x=72.5\) and above \(y=52.5\); \(A \cup B\) includes left of \(x=72.5\) or above \(y=52.5\).

Step 1: Define the Sample Space

The sample space for the rise time of two batches can be represented as a two-dimensional plane, where the x-axis corresponds to the rise time of batch 1 and the y-axis corresponds to the rise time of batch 2. The sample space includes all possible combinations of positive real numbers for both batches. This is shown as the first quadrant of the Cartesian plane, since rise times are positive.

Step 2: Represent Event A on the Graph

Event \(A\) represents the condition where the rise time of batch 1 is less than 72.5 minutes. On the two-dimensional plot, this corresponds to the region where the x-values are less than 72.5, which is the area to the left of the vertical line at \(x = 72.5\). Shade this region on the graph.

Step 3: Represent Event B' on the Graph

Event \(B'\) (the complement of event \(B\)) represents the condition where the rise time of batch 2 is less than or equal to 52.5 minutes. On the graph, this corresponds to the region where the y-values are less than or equal to 52.5, which is the area below the horizontal line at \(y = 52.5\). Shade this area on the graph.

Step 4: Represent Event A ∩ B on the Graph

Event \(A \cap B\) represents the intersection of events \(A\) and \(B\). This is where both conditions are satisfied: the rise time of batch 1 is less than 72.5 minutes, and the rise time of batch 2 is greater than 52.5 minutes. On the graph, this corresponds to the area to the left of the line \(x = 72.5\) and above the line \(y = 52.5\). Shade this intersection region.

Step 5: Represent Event A ∪ B on the Graph

Event \(A \cup B\) represents the union of events \(A\) and \(B\). This includes any region where either event \(A\) is true, or event \(B\) is true (or both). Thus, it includes the area to the left of \(x = 72.5\), the area above \(y = 52.5\), or both. Shade the entire region that satisfies at least one of these conditions on the graph.

Key Concepts

Sample Space

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.

Two-Dimensional Plot

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

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.

Intersection and Union of Events

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.