# 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

### Concepts

Kolmogorov's Axioms

### Examples

Directions: The following examples cover the material from the video above.

### 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.
1. What is a probability space? Can you state the axioms?
2. What is the difference between an outcome and an event in a probabilistic model? To which one do we assign probability?
3. Does $P\left(A\right)=0$ imply $A=\mathrm{\varnothing }$? Can you provide a counterexample, if not?
4. Can you describe and $\bigcap _{n=1}^{\mathrm{\infty }}\left(-\frac{1}{n},\frac{2}{n}\right)$?
5. Does $\phantom{\rule{1em}{0ex}}\cdot \phantom{\rule{-10pt}{0ex}}\bigcup _{n=1}^{\mathrm{\infty }}\left[\frac{1}{n},1\right]$ make sense?
Discrete Uniform Probability Spaces

### Examples

Directions: The following examples cover the material from the video above.

### 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.
1. Can we have an infinite discrete uniform probability space?
2. 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: The following examples cover the material from the video above.

### 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.
1. What does the Complement Rule say, and why is it useful?
2. 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: The following examples cover the material from the video above.

### 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.
1. State the monotonic property of probability.
2. When do we have $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$?