Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Are the events \(E\) and \(F\) mutually exclusive?

Step-by-Step Solution

Answer

No, the events E and F are not mutually exclusive, since their intersection is not empty: Intersection(E, F) = {a}.

Step 1: Identify the events E and F

Event E = {a, b} Event F = {a, d, f}

Step 2: Check for the intersection of events E and F

We will now determine the intersection of E and F. The intersection of two sets is the set of elements that are common to both sets. Intersection(E, F) = {a}

Step 3: Determine if the events E and F are mutually exclusive

Since the intersection of events E and F contains an element, a, the events E and F are not mutually exclusive.

Key Concepts

Sample Space

In probability, a **sample space** is a fundamental concept. It's like the foundation of any probability problem. The sample space is the set of all possible outcomes of an experiment. In our exercise example, we are given the sample space \( S = \{a, b, c, d, e, f\} \).

This means that when we conduct the experiment, every result we get will be one of these elements in the set. It's crucial to identify this at the beginning of any problem because all probability calculations will be based on these possibilities.

• Ensure all possible outcomes are considered.
• Each event like \(E\), \(F\), and \(G\) is a subset of this sample space.
• Think of the sample space as a map for the event outcomes to exist within.

Intersection of Sets

Understanding the **intersection of sets** is key when working with probabilities. The intersection refers to the elements common to both sets. In mathematical terms, if \(E\) and \(F\) are subsets of our sample space, then the intersection \(E \cap F\) is the set of elements they share.

For instance, in our exercise:

• Event \(E = \{a, b\}\)
• Event \(F = \{a, d, f\}\)

When we find \(E \cap F\), we look for common elements in both sets \(E\) and \(F\). The result here is \(\{a\}\), indicating that "a" is the only outcome common to both events. This concept helps us determine the nature of the relationship between the two events.

Understanding intersections is vital for distinguishing whether events can happen simultaneously or not.

Probability Concepts

At the heart of this problem is the idea of **probability concepts**, which involves determining how likely events are to occur. Here, we focus on "mutually exclusive events."

Mutually exclusive events are those that cannot happen at the same time. In simpler terms, no outcome can belong to both events. If their intersection has no elements (is an empty set), we call them mutually exclusive. For events \(E\) and \(F\), when we calculated the intersection: \(E \cap F = \{a\}\), we observed that it was not empty because the element "a" was common in both events.

• This indicates \(E\) and \(F\) are not mutually exclusive.
• The presence of any element in the intersection negates mutual exclusivity.

This understanding is crucial in probability as it influences how we calculate combined probabilities of events.