Problem 11 - Algebra 2 and Trigonometry

Question

For the sample space \(\\{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\\},\) determine how many events are possible.

Step-by-Step Solution

Answer

There are 16 possible events.

Step 1: Understand What is an Event

In probability and statistics, an event is a subset of the sample space. Each event represents a possible outcome or a group of outcomes from an experiment.

Step 2: Identify the Sample Space

The sample space for this exercise is given by \( \{ \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D} \} \). It consists of 4 elements.

Step 3: Determine the Number of Subsets

The number of events (subsets) possible in a sample space with \( n \) elements is calculated using the formula \( 2^n \). Here, \( n = 4 \) since the sample space consists of 4 elements.

Step 4: Apply the Formula

Calculate \( 2^4 \) to find the total number of events. \( 2^4 = 16 \).

Step 5: Verify the Answer

List a few subsets to verify understanding: 1. An empty set \( \{ \} \) is a subset.2. Single element subsets such as \( \{ \mathrm{A} \} \), \( \{ \mathrm{B} \} \), etc.3. Two element subsets such as \( \{ \mathrm{A}, \mathrm{B} \} \), \( \{ \mathrm{B}, \mathrm{C} \} \), etc.Thus confirming there are indeed 16 possible subsets.

Key Concepts

Sample Space: The Foundation of Probability

In probability, the sample space is crucial. It is the set of all possible outcomes of a particular experiment or random event. Imagine rolling a die; the sample space would include numbers from 1 to 6.
In the example provided in the exercise, the sample space is given as \( \{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\} \). But why is sample space so important? It represents every potential outcome.

• Helps identify the events we can calculate probabilities for.
• Establishes the universe within which all probability calculations are made.
• Serves as the starting point for listing all possible events and analyzing them.

Each element in the sample space is a single outcome from the experiment. Properly understanding the sample space is the first step in tackling any probability problem.

Subsets: Building Blocks of Events

Subsets are groups of elements taken from the sample space. Each subset corresponds to an event in probability. Let's return to our sample space \( \{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\} \).

• The empty set \( \{\} \) is a subset, representing no occurrence.
• Single element subsets like \( \{\mathrm{A}\} \) or \( \{\mathrm{B}\} \) represent individual events.
• Multiple element subsets, such as \( \{\mathrm{A}, \mathrm{B}\} \), represent events where these elements occur together.

Each subset is like a tiny universe on its own, capturing specific outcomes. In probability, every subset is a potential event, whether it holds all items from the sample space or none.

Number of Events: Calculating Possibilities

Determining the number of possible events describes the range of outcomes we can account for. In any sample space with \( n \) elements, the number of potential events (or subsets) is given by the formula \( 2^n \).
For the sample space \( \{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\} \), this translates to \( 2^4 \), equating to 16 distinct events.

• One way to look at it: each element either participates in a subset or it doesn't, forming a binary decision.
• This leads to exponential growth in possibilities as more elements enter the sample space.
• Listing a few demonstrates this: the empty set, single elements like \( \{\mathrm{A}\} \), two-element combinations like \( \{\mathrm{A}, \mathrm{B}\} \), all the way up to the full set itself.

These calculations form the backbone of establishing event possibilities, guiding the understanding of probability and its diverse outcomes.