# Probability Model The core concept in probability theory is that of a **probability model**. It consists of three components: 1. **Sample Space**: The sample space is a *set* whose elements are called *outcomes* or *sample points*. 2. **Events**: Class of all subsets of the sample space. Note that an event is made up of greater than or equal to 1 outcome (i.e. n-tuples). 3. **Probability Measure**: Assignment of a nonnegative number to each outcome, with the restriction that these numbers must sum to one over the sample space. The probability of an event is the sum of the probabilities of the outcomes comprising that event. This is referred to as a **Probability Law** in certain texts. An example of a probability model for a die roll can be seen below:  ### Observable probability vs real world frequency  ### Single roll model vs. n roll model When rolling a die a single time the sample space is simply $\{ 1,2,3,4,5,6\}$ and we are dealing with 1-tuples, hence our sample space has $6^1$ outcomes. However, in the case of $n$ rolls, we have $6^n$ possible outcomes. Hence, our sample space have $6^n$ outcomes. The key is to keep in mind that these are *different* experiments/probability models. We have a single roll model and an $n$ roll model. The distinction here is important because it influences what we mean by "sample point". If we are working with a single roll model, our sample points are 1-tuples. If we are working with an $n$ roll model, our samples (outcomes) are n-tuples. ### Outcome vs Event  ### Idealized Experiment vs. Probability Model  ### Repeated Idealized experiments Much of our intuitive understanding of probability comes from the notion of repeating the same idealized experiment many times. However, the axioms of probability contain no explicit recognition of such repetitions. The appropriate way to handle n repetitions of an idealized experiment is through an extended experiment whose sample points are n-tuples of sample points from the original experiment. Such an extended experiment is viewed as n trials of the original experiment. The notion of multiple trials of a given experiment is so common that one sometimes fails to distinguish between the original experiment and an extended experiment with multiple trials of the original experiment. This can be handled mathematically via the [cartesian-product](cartesian-product.md). ### Building a probability model for the real world How do you build a probability model for the real world? There are two answers. Let us start with the decent answer: * Learn about estimation and decisions within standard models, i.e. within a completely mathematical framework. Then learn a great deal about the real world problem. Then use common sense and tread lightly. Now here is the better answer: > Try over simplified models first. Use the mathematics of those simple models to help understand the real problem. Then, in multiple stages, add to and modify the models to understand the original problem better. This falls in line with the general approach of *layering complexity*, and the even more general approach of *conjecture and criticism*, or *conjecture and error correction*. Never forget, [Intelligence is Error Correction](Intelligence%20is%20Error%20Correction.md). --- Date: 20211007 Links to: [Probability MOC](Probability%20MOC.md) Tags: References: * Discrete Stochastic Processes Notes, Gallagher - Fantastic resource! ([here](https://drive.google.com/file/d/1DsCW0L8lLt6YdF2SBNK73UJh1oUcerwQ/view?usp=sharing))