Problem 29 - Applied Statistics and Probability for Engineers

Question

The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events \(A\) and \(B\) as follows: \(A=\\{x \mid x<72.5\\}\) and \(B=\\{x \mid x>52.5\\}\) Describe each of the following events: (a) \(A^{\prime}\) (b) \(B^{\prime}\) (c) \(A \cap B\) (d) \(A \cup B\)

Step-by-Step Solution

Answer

(a) \(A'=\{x \mid x \geq 72.5\}\); (b) \(B'=\{x \mid x \leq 52.5\}\); (c) \(A \cap B=\{x \mid 52.5 < x < 72.5\}\); (d) \(A \cup B=\{x \mid x \neq 52.5\}\).

Step 1: Understanding Event A

Event \(A\) is defined as \(A=\{x \mid x<72.5\}\). This means that \(A\) includes all positive real numbers less than 72.5 minutes.

Step 2: Understanding Event A' (Compliment)

Event \(A'\), or the compliment of \(A\), is defined as all outcomes not in \(A\). Therefore, \(A'=\{x \mid x \geq 72.5\}\), meaning all positive real numbers greater than or equal to 72.5 minutes.

Step 3: Understanding Event B

Event \(B\) is defined as \(B=\{x \mid x>52.5\}\). This means that \(B\) includes all positive real numbers greater than 52.5 minutes.

Step 4: Understanding Event B' (Compliment)

Event \(B'\), or the compliment of \(B\), is defined as all outcomes not in \(B\). Therefore, \(B'=\{x \mid x \leq 52.5\}\), meaning all positive real numbers less than or equal to 52.5 minutes.

Step 5: Finding Event A \\u2229 B (Intersection)

The intersection \(A \cap B\) includes all numbers that are in both \(A\) and \(B\). Since \(A=\{x \mid x<72.5\}\) and \(B=\{x \mid x>52.5\}\), \(A \cap B = \{x \mid 52.5 < x < 72.5\}\). This describes all positive real numbers between 52.5 and 72.5 minutes.

Step 6: Finding Event A \\u222a B (Union)

The union \(A \cup B\) includes all numbers that are in \(A\), \(B\), or in both. Since \(A=\{x \mid x<72.5\}\) and \(B=\{x \mid x>52.5\}\), \(A \cup B = \{x \mid x < 72.5\} \cup \{x \mid x > 52.5\}\). This describes all positive real numbers except for 72.5 and lower values not greater than 52.5.

Key Concepts

Sample Space

A sample space in probability theory is essentially the complete set of all possible outcomes of an experiment. In the given exercise, the sample space is defined as positive real numbers. These are numbers greater than zero that can have decimal points, meaning fractions of a minute are possible and allowed.

For example, if we are measuring the rise time of a reactor, the sample space might include all numbers from just above 0 up to larger numbers, like 100 or 200 minutes.
This range shows that if the rise time is 0.1, 15.75, or 100.1 minutes, they're all valid outcomes.

• Positive numbers only — zero and negative numbers are not considered.
• Includes fractions and whole numbers.

Understanding this concept helps in framing what possible outcomes can be expected when we measure the rise time.

Event Complement

In probability, the complement of an event is a central concept. It refers to everything in the sample space that is not included in the event being considered. For example, if event \(A\) includes rise times less than 72.5 minutes, the complement \(A'\) would include times 72.5 minutes or greater.

Why is this useful? Knowing the complement helps in situations where finding the probability of the original event directly might be complex. Instead, you calculate the probability of its complement and subtract from 1, because the sum of probabilities of an event and its complement is always 1.

• For event \(A\), its complement \(A' = \{x \mid x \geq 72.5\}\)
• For event \(B\), its complement \(B' = \{x \mid x \leq 52.5\}\)

Always remember: the complement includes all other possible outcomes within the sample space.

Intersection and Union in Probability

The intersection of two sets in probability, noted as \(A \cap B\), represents the set of elements that are members of both sets. In our exercise, \(A \cap B = \{x \mid 52.5 < x < 72.5\}\). This represents rise times that are both greater than 52.5 minutes and less than 72.5 minutes.

On the other hand, the union, noted as \(A \cup B\), represents the set of elements that are members of either set or both. In the given problem, \(A \cup B = \{x \mid x < 72.5\} \cup \{x \mid x > 52.5\}\). This means it includes all rise times, except times exactly equal to 52.5 minutes.

• Intersection \(A \cap B\): both conditions must be true simultaneously.
• Union \(A \cup B\): either one or both conditions must be true.

These concepts are fundamental for determining combined probabilities and for understanding how overlapping and disjoint sets function in probability.

Positive Real Numbers

Positive real numbers are a type of number in mathematics that includes all possible values greater than zero. They feature prominently in probability problems as sample spaces often include variables that require positive measures, like time or distance.

In the context of our exercise, since we're dealing with time, negative values or zero aren’t applicable. It’s crucial to consider that:

• Numbers can be both whole numbers (like 3, 100) and decimal numbers (like 0.01, 52.78).
• They extend indefinitely, meaning there is no upper limit in theory.

Understanding positive real numbers is essential when working with probability, as they define the numerical outcomes that you can realistically expect from an experimental or observational situation.