Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S\) be any sample space and let E, \(\boldsymbol{F}\), and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(\cup, \cap\), and . The event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs

Step-by-Step Solution

Answer

The event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs can be represented using the symbols \(\cup\), \(\cap\), and \(\complement\) as follows: \(E \cap \left(F^\complement \cap G^\complement\right)\)

Step 1: Identify the Symbols

Before we start, let's recall the meaning of these symbols: 1. \(\cup\): The union symbol denotes the event where either of the events (or both) occur. Mathematically, \(A \cup B\) represents the event where either \(A\) or \(B\) (or both) occurs. 2. \(\cap\): The intersection symbol denotes the event where both events occur simultaneously. Mathematically, \(A \cap B\) represents the event where both \(A\) and \(B\) occur. 3. \(\complement\): The complement symbol denotes the event where the event does NOT occur. Mathematically, \(A^\complement\) represents the event where \(A\) does not occur. We will use these symbols to describe the event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs.

Step 2: Working with Events

Our goal is to describe the event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs. First, let's break this down into two separate conditions: 1. \(E\) occurs: This is simply event \(E\). 2. Neither \(F\) nor \(G\) occurs: To describe this, we will use the complement symbol, denoting that both \(F\) and \(G\) do not occur. Mathematically, this is represented as \(F^\complement \cap G^\complement\) (i.e., both \(F\) and \(G\) do not occur).

Step 3: Combining the Conditions

Now, we need to combine these conditions to describe the event we're looking for. Since we want \(E\) to occur AND neither \(F\) nor \(G\) to occur, we can represent this by taking the intersection of these two events: \(E \cap \left(F^\complement \cap G^\complement\right)\) And that's our answer! This expression represents the event where \(E\) occurs but neither of the events \(F\) or \(G\) occurs.