# 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: You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
1. Suppose you have a biased coin in which heads is 3 times more likely to occur than tails. If you flip this coin, what is the probability of getting heads? tails?

$$P(\{H\})=\dfrac{3}{4} \mbox{ and } P(\{T\})=\dfrac{1}{4}$$

2. Flip a fair coin until a tail comes up. Establish the probabilistic model for such a random
experiment.

See video for further 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.
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: 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?

$$\dfrac{1}{12}$$

### 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: 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?

$$\dfrac{13}{18}$$

### 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: You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
1. 15% of the population in a town is blond, 25% has blue eyes, and 2% is blond with blue eyes. What is the probability that a randomly chosen individual is not blond and does not have blue eyes?

2. There are 30 red balls, 20 green, and 10 yellow balls in an urn. Draw 7 balls without replacement. What is the probability that exactly 2 red or exactly 3 yellow balls are in the sample?

$$P(A\cup B) = \displaystyle \frac{ {30 \choose 2 } {30 \choose 5}}{{60 \choose 7}}+ \frac{\binom{10}{3} \binom{50}{4}}{\binom{60}{7}} - \frac{\binom{30}{2} \binom{10}{3} \binom{20}{2}}{\binom{60}{7}}$$

### 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 $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$?