Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\).

Step-by-Step Solution

Answer

The statement is true. Since all elements of \(A\) are also elements of \(B\), the probability of \(A\) occurring must be less than or equal to the probability of \(B\) occurring. Mathematically, this is shown as \(P(A) \leq P(B)\).

Step 1: Understanding subsets and probability

A subset is a set containing elements that are all included in another set. In this problem, \(A\) is a subset of \(B\), which means all elements in \(A\) are also elements in \(B\). Probability is the measure of the likelihood of an event occurring. If \(A\) is an event, \(P(A)\) represents the probability of that event happening. Similarly, \(P(B)\) represents the probability of event \(B\) happening. In this problem, we need to compare the probabilities of events \(A\) and \(B\) when \(A\) is a subset of \(B\).

Step 2: Comparing probabilities of subsets

As \(A\) is a subset of \(B\), we know all elements in \(A\) are also elements in \(B\). Therefore, if there's any probability of an element in \(A\) occurring, this probability must also be included in the probability of \(B\) occurring. This is because \(B\) must account for all its elements and the likelihood of those elements occurring, including those from \(A\). So, the probability of \(A\) occurring must be less than or equal to the probability of \(B\) occurring, since \(B\) contains all the elements of \(A\) as well as its own elements. Mathematically, this means \(P(A) \leq P(B)\).

Step 3: Verifying the statement is true

We have shown that if \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\), based on the fact that all elements in \(A\) are also elements in \(B\). So the given statement is true. To support this, we can provide a brief explanation. Explanation: Since all elements of \(A\) are also elements of \(B\), the probability of \(A\) occurring is either a part of or equal to the probability of \(B\) occurring. Therefore, the probability of event \(A\) must be less than or equal to the probability of event \(B\), which is represented mathematically as \(P(A) \leq P(B)\).

Key Concepts

Subsets

The concept of subsets is pivotal in understanding relationships between different sets. A set is a collection of distinct objects or elements. When we say that set \(A\) is a subset of set \(B\), we mean every element of \(A\) is also an element of \(B\). This relationship is written as \(A \subseteq B\).
For example, consider a set \(B\) containing elements \{1, 2, 3\}, and a set \(A\) containing elements \{1, 2\}. Here, \(A\) includes only elements found in \(B\), so \(A\) is a subset of \(B\).

• If \(A = \{a, b, c\}\) and \(B = \{a, b, c, d\}\), then \(A \subseteq B\) as all elements of \(A\) exist in \(B\).
• Two sets that are equal can also be thought of as each being a subset of the other; for instance, if \(A = B\), then both \(A \subseteq B\) and \(B \subseteq A\).

Understanding subsets is essential for working with probabilities in various contexts, as it provides a foundation for comparing the likelihood of different outcomes.

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are fundamental objects in mathematics. A set is simply a collection of distinct objects, and the essential operations we perform with sets include union, intersection, and complementation. These operations define how sets interact with each other and are crucial for problem-solving and reasoning.

• Union: The union of two sets \(A\) and \(B\), denoted by \(A \cup B\), is the set containing all elements from both \(A\) and \(B\).
• Intersection: The intersection of two sets \(A\) and \(B\), denoted by \(A \cap B\), includes only the elements that are common to both \(A\) and \(B\).
• Complement: The complement of a set \(A\), relative to a universal set \(U\), is the set of elements in \(U\) that are not in \(A\), denoted by \(A^c\).

In probability, set theory helps define relationships among different events. For instance, when we talk about the probability of either event \(A\) or \(B\) occurring, we use the union operation. Similarly, when considering the simultaneous occurrence of events, we use the intersection.
Understanding these operations enhances our ability to manipulate and comprehend the probabilities of complex events.

Probability Comparison

In probability theory, comparing event probabilities is crucial for understanding the likelihood of different outcomes and making informed decisions. When dealing with subsets, like event \(A\) being a subset of event \(B\), it’s important to compare their probabilities.
When \(A \subseteq B\), every possible outcome in event \(A\) is also in event \(B\). Therefore, it follows that the probability of event \(A\), denoted as \(P(A)\), must be less than or equal to the probability of event \(B\), \(P(B)\). This implies:

• \(P(A) \leq P(B)\) because every outcome that makes \(A\) happen also contributes to \(B\) happening.
• The probability of \(B\) can be greater because it might include other outcomes in addition to those in \(A\).

To understand this intuitively, imagine probability as a container of possibilities. If \(A\) is inside \(B\), all water (probability) filling \(A\) also fills \(B\), but \(B\) may hold more since it's larger.
By grasping this concept, students can better analyze situations where one event is entirely within another, predicting outcomes and making calculations more effectively through probability comparisons.