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.
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?
2. Flip a fair coin until a tail comes up. Establish the probabilistic model for such a random experiment.
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.
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?
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 \(P(A\cup B)=P(A)+P(B)?\)