Problem 4 - Algebra and Trigonometry

Question

True or False. In a probability model, the sum of all probabilities is \(1 .\)

Step-by-Step Solution

Answer

True. The sum of all probabilities in a probability model equals 1.

Step 1: Understanding the Problem

The question asks whether, in a probability model, the sum of all probabilities equals 1. This means we need to examine the basic properties of probability to determine the correctness.

Step 2: Review Probability Basics

In a probability model, each outcome of a sample space is assigned a probability. Probabilities are values between 0 and 1, inclusive.

Step 3: Sum of Probabilities in a Sample Space

One of the key axioms of probability is that the sum of the probabilities of all mutually exclusive outcomes in a sample space must equal 1. This is a fundamental principle in probability theory.

Step 4: Conclusion

Given the reviewed probability principles, the statement that in a probability model, the sum of all probabilities is 1, is indeed correct.

Key Concepts

Probability Model

A probability model is a mathematical representation of a random phenomenon. It consists of a sample space, events formed from the sample space, and a probability assigned to each event.
A probability model has two main components:

• The sample space, denoted by \(S\), which includes all possible outcomes of the experiment.
• A probability function, \(P\), that assigns a probability to each event in the sample space.

The probabilities assigned must adhere to certain rules or axioms, ensuring they provide a coherent and consistent description of the random phenomenon.

Sample Space

The sample space is the set of all possible outcomes of a probability experiment.
For example, if you are rolling a six-sided die, the sample space is:
\[S = \{1, 2, 3, 4, 5, 6\}\]
Each outcome in the sample space is equally likely if the die is fair. The sum of the probabilities of all individual outcomes must equal 1. The events, which are subsets of the sample space, are typically what we are interested in. An event could be 'rolling an even number' like \(\{2, 4, 6\}\).

Axioms of Probability

The axioms of probability are the foundational rules that all probabilities must follow. They ensure coherency and provide a consistent framework for probability analysis:

• Non-negativity: The probability of any event is a non-negative number: \(P(E) \geq 0\) for any event \(E\).
• Normalization: The sum of the probabilities of all mutually exclusive outcomes must equal 1: \[ \sum_{i=1}^{n} P(E_i) = 1 \]
• Additivity: If \(E_1\) and \(E_2\) are mutually exclusive events, then the probability of their union is the sum of their probabilities: \(P(E_1 \cup E_2) = P(E_1) + P(E_2)\).

These axioms ensure that the structure and calculation of probabilities remain consistent and reliable.