# Section 1: Probabilistic Models and Probability Laws

### Instructions

- This section covers the concepts listed below.
- For each concept, there is a conceptual video explaining it followed by videos working through examples.
- When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page.

## Kolmogorov's Axioms

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

### Self-Assessment Questions

**Directions:**The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.

- What is a probability space? Can you state the axioms?
- What is the difference between an outcome and an event in a probabilistic model? To which one do we assign probability?
- Does \(P(A)=0\) imply \(A=\emptyset\)? Can you provide a counterexample, if not?
- Can you describe \(\displaystyle \bigcup_{n=1}^{\infty}\left[\frac{1}{n},1\right]\) and \(\displaystyle \bigcap_{n=1}^{\infty}\left(-\frac{1}{n},\frac{2}{n}\right)\)?
- Does \(\displaystyle\cdot \hspace{-11pt}\bigcup_{n=1}^{\infty} \left[\frac{1}{n},1\right]\) make sense?

## Discrete Uniform Probability Spaces

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

1. We roll a pair of fair dice. What is the probability of getting a sum of 10?

### Self-Assessment Questions

**Directions:**The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.

- Can we have an infinite discrete uniform probability space?
- Does "rolling a pair of fair dice and recording the product of the two numbers as the outcome" define a uniform probability space?

## Complement Rule

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

1. Roll a fair die four times. What is the probability that some number appears more than once?

### Self-Assessment Questions

**Directions:**The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.

- What does the Complement Rule say, and why is it useful?
- What is the complement of the following event in the random experience of rolling two dice: \(A=\) the event of getting a sum of at least 4? Determine all the outcomes of \(A^c.\)

## Monotonic Property and Inclusion & Exclusion

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

### Self-Assessment Questions

**Directions:**The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.

- State the monotonic property of probability.
- When do we have \(P(A\cup B)=P(A)+P(B)?\)