Problem 7 - Algebra and Trigonometry

Question

Determine whether the following is a probability model. $$\begin{array}{cc}\text { Outcome } & \text { Probability } \\\\\hline 1 & 0.2 \\\2 & 0.3 \\\3 & 0.1 \\\4 & 0.4\end{array}$$

Step-by-Step Solution

Answer

Yes, it is a valid probability model.

Step 1: Understand the Definition of a Probability Model

A probability model must satisfy two main conditions: 1. Each probability must be between 0 and 1, inclusive. 2. The sum of all the probabilities must equal 1.

Step 2: Check Individual Probabilities

Inspect each given probability to ensure it falls between 0 and 1. For this exercise, we have the probabilities: 0.2, 0.3, 0.1, and 0.4. Each value is within the range [0, 1], so this condition is satisfied.

Step 3: Sum the Probabilities

Add all the probabilities together to determine if their sum equals 1. Calculate: \[ 0.2 + 0.3 + 0.1 + 0.4 = 1.0 \]

Step 4: Conclusion

Since both conditions are satisfied (individual probabilities are between 0 and 1, and their sum equals 1), this is a valid probability model.

Key Concepts

Probability Definition

In probability theory, probability refers to the likelihood of an event happening. It is usually denoted by a number between 0 and 1, where 0 means the event cannot happen at all, and 1 means the event is certain to happen. Each outcome in a probability model has a probability assigned to it, representing the chance of that specific outcome occurring.
For example, consider the outcome of rolling a die. Each face (from 1 to 6) has a probability of \(\frac{1}{6}\) since a standard die has six faces, and each face is equally likely to land face up.

Sum of Probabilities

One critical aspect of probability models is that the sum of all assigned probabilities must equal 1. This fundamental rule ensures that one of the many possible outcomes will indeed occur.
In the given exercise, we check if this rule holds by adding the probabilities of all outcomes. The probabilities given are 0.2, 0.3, 0.1, and 0.4.
Calculating the sum: \[ 0.2 + 0.3 + 0.1 + 0.4 = 1.0 \]
This confirms that the sum of the probabilities is 1, fulfilling this crucial requirement of a probability model.

Valid Probability Range

Each probability in a model must fall within the range of 0 to 1, inclusive. This ensures that the likelihood of an event is logically sound, as negative probabilities or probabilities greater than 1 are not feasible.
Let’s examine the probabilities in the exercise: 0.2, 0.3, 0.1, and 0.4. Each of these values is between 0 and 1.
Thus, they all satisfy the valid probability range requirement.
In summary, a probability model needs to meet two conditions: 1) All probabilities are between 0 and 1, and 2) The sum of all probabilities is 1. This exercise confirms that both conditions are satisfied, making it a valid probability model.