Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. If the probability of an outcome in a sample space is 1 then the probability of the other outcomes in the sample space is \(0 .\)

Step-by-Step Solution

Answer

The statement is true. If the probability of an outcome in a sample space is 1, then the probability of the other outcomes in the sample space is indeed 0.

Step 1: Understand the Definition of Probability

The probability of an event happening or not can be determined between 0 and 1. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain to occur.

Step 2: Understand the Rules of Probability

One important rule to remember is that all probabilities must add up to 1 in a sample space. Therefore, if the probability of an event A is given as 1, it implies that the event is certain to occur in the sample space.

Step 3: Analyze the given statement

The statement says 'If the probability of an outcome in the sample space is 1 then the probability of the other outcomes in the sample space is 0'. Based on our understanding of probability, this statement is true. When the probability of an outcome is 1, it means the event is certain to occur and since all probabilities in a sample space add up to 1, there is no probability left for the other outcomes, hence their probability is 0.

Key Concepts

Sample Space

In probability theory, the concept of a sample space is fundamental. It refers to the set of all possible outcomes of a random experiment. For example, when tossing a fair six-sided die, the sample space consists of the outcomes \( \{1, 2, 3, 4, 5, 6\} \). Each of these outcomes is a distinct possibility resulting from the experiment.
Understanding the sample space is crucial because it forms the foundation upon which all probability calculations are based. By clearly defining the sample space, we ensure that we accurately assess the likelihood of various outcomes.

• The sample space provides the overall context for analyzing probability.
• It ensures we assess all possible outcomes.
• Clear identification of the sample space avoids errors in probability computations.

Outcome Probability

The probability of an outcome describes how likely that outcome is, within a given sample space. For each possible outcome, we assign a probability value. This value is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. An outcome's probability is calculated by considering the favorable outcomes over the total outcomes in the sample space.
For a simple example, consider our six-sided die again. With an unbiased die, the probability of rolling a 4 is \(\frac{1}{6}\), since there is only one favorable outcome (rolling a 4) out of six possible outcomes.

• Outcome probabilities help determine the likelihood of specific events.
• They must add up to 1 within the entire sample space.
• Assigning probabilities is essential for precise prediction and risk assessments.

Probability Rule

Probability rules are a set of principles governing how probabilities are assigned and interact within a sample space. One key rule is that the sum of probabilities of all possible outcomes in a sample space must equal 1. This is because probability measures certainty, and all outcomes together encompass certainty within an experiment.
For example, if we know the probability of an event within a sample space is 1, it means the event is certain, leaving no room for other events to occur. Therefore, the probability of all other outcomes is 0.

• The sum of probabilities must be exactly 1.
• If an outcome is certain (probability = 1), others must be impossible (probability = 0).
• Probability rules ensure consistent and logical outcomes.

Understanding these rules forms the backbone of correctly interpreting probability scenarios.