Problem 1 - Algebra and Trigonometry

Question

The set of all possible outcomes of an experiment is called the _____ _____ A subset of the sample space is called an ____.

Step-by-Step Solution

Answer

Sample space; Event.

Step 1: Define Sample Space

The set of all possible outcomes of an experiment is known as the sample space. It includes every possible scenario that could occur in the experiment. For example, when rolling a die, the sample space is the set {1, 2, 3, 4, 5, 6}.

Step 2: Define Event

An event is a subset of the sample space. It includes one or more outcomes from the sample space. For example, if we consider the event of rolling an even number on a die, the subset of the sample space could be {2, 4, 6}.

Key Concepts

Event

In probability theory, an event is a significant concept that refers to a subset of the sample space. This means it is a specific group of outcomes that we are interested in when performing an experiment. Imagine rolling a six-sided die. The sample space of this die is {1, 2, 3, 4, 5, 6}, meaning these are all the possible outcomes we can get.

Now, if we want to focus on an event, such as rolling an even number, we look at this specific subset of outcomes. Here, the event is:

• Rolling an even number: {2, 4, 6}

This event represents a portion of all possible outcomes, specifically those that satisfy our condition of being even.

Events can vary in complexity. They range from single outcomes to combinations of several outcomes. For example, the event of rolling a number less than 4 can be expressed as the subset {1, 2, 3} from the sample space. Being clear about what constitutes an event is essential when calculating probabilities.

Probability

Probability is the measure of how likely an event is to occur within a given sample space. It's expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 signifies a certainty. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space.

Here's the basic formula for probability:

• Probability = (Number of favorable outcomes) / (Total number of outcomes)

Let's consider our earlier example of rolling a die. If the event is rolling an even number, the favorable outcomes are {2, 4, 6}, and the total number of outcomes is 6 (since a die has six faces). Therefore, the probability is:

• Probability of rolling an even number = 3/6 = 0.5 or 50%

Understanding probability helps us predict the likelihood of future events based on known conditions. In real-world applications, this concept assists in decision making, risk assessment, and statistical analysis.

Subset

A subset is a fundamental idea in set theory and an essential component in understanding events within a sample space. When you have a set, a subset is a collection of elements from that set, including the possibility of it being the set itself or even an empty set.

Let's break down the concept using our die rolling example. The sample space is {1, 2, 3, 4, 5, 6}. Subsets of this sample space could include:

• {1, 2}
• {4, 5, 6}
• {}
• {1, 2, 3, 4, 5, 6}

The subset

• {2, 4, 6}

was specifically used in earlier sections as it represents the outcomes of the event: rolling an even number.

Subsets are powerful because they allow us to narrow down focus and make calculations or create conditions specific to certain scenarios. They serve as building blocks in both mathematical theories and practical probability applications.