A Wolfram U Daily Study Group on Introduction to Probability begins on February 27th 2023.
Join me and a group of fellow learners to learn about the world of probability and statistics using the Wolfram Language. Our topics for the study group include the characterisation of randomness, random variable design and analysis, important random distributions and their applications, probability-based data science and advanced probability distributions.
The idea behind this study group is to rapidly develop an intuitive understanding of probability for a college student, professional or interested hobbyist. A basic working knowledge of the Wolfram Language is recommended but not necessary. We are happy to help beginners get up to speed with Wolfram Language using resources already available on Wolfram U.
Please feel free to use this thread to collaborate and share ideas, materials and links to other resources with fellow learners.
The study group starts next week, on February 27th.
If you want to take full advantage of this course’s material and get a practical and deep understanding of probability, don’t forget to click on the REGISTER HERE link to get registered in this course!
I’m looking forward to your participation and feedback!
Reminder that our upcoming Daily Study Group provides a preview of the new interactive course, Introduction to Probability. The Study Group meets daily over two weeks, Monday through Friday, for an hour online each day, starting Monday. Take advantage of this opportunity to prepare for probability and statistics related coursework and research in natural science, engineering, finance, medicine, data science and other fields! You can sign up here.
This course will fit well with my university courses of Applied Probability and Statistics STA 345 and Probability and Statistics I STA 445 and Probability and Statistics II 446.
I think applying probability theory to Catan is interesting. For example there are two dice in Catan with six sides and the most common roll is a 7 because the expectation of a normal dice is 3.5 and 3.5+3.5=7. The 7 causes the robber to make you discard half of your resource and commodity cards rounded down if you have more than the discard limit.
I would like to mention that you can calculate the combinatorial enumeration of all events in the sample space with FactorialPower. For example, with the example from the second presentation
Imagine that 4 Swiss and 5 Ethiopian athletes compete for the 200m sprint.
How many different top-three winner rankings are possible?
FactorialPower[9,3] returns 504 which is the same as 9!/(9-3)!.
Emails have just now been sent to everyone who is registered for the Daily Study Group. Recordings can be accessed from the webinar series landing page.
The probabilistic statements of quantum mechanics are a useful application of probability theory. This is not approached in this course not because of its difficulty, but because of necessary background knowledge of quantum physics to be able to use those statements. However, the probability of decay is mentioned Lesson and Exercises 17, as it is a classical example for exponential distributions.
Yes, as said in the beginning, this covers the material for Probability courses. However, Statistics are not covered, you will need your courses for that.
As for the probabilities of throwing two dice, there is far more knowledge needed to formulate this than meets the eye. As said in the introduction, the probabilities of dice are explored throughout the lessons. I believe if you seek the exact probabilistic formulation of this problem, you can look at Lesson 22.
Indeed! FactorialPower can be used, you’re welcome to do that. However, the goal of Lesson 2 was to get a good understanding of combinatorics, so it seemed more intuitive to just do it explicitly.
I downloaded your class notebooks from the site today.
Slide 7 of lesson 2 “Consider that the 4 Swiss and 5 Ethiopian athletes want to form a team of 5 to represent them in competition. How many different teams are possible given that it must include 3 Swiss and 2 Ethiopians?” Shouldn’t the answer be Binomial[4,3]*Binomial[5,2]?
I am a high school math teacher. The lesson materials are wonderful to be used in our classroom. There are some simulations in the notebook of day 1. I wonder is there any place that I can download the source code? just like the “normal convergence” you shared.
Hi…two fast questions:
1- In these two last laws (pls. see the image) .. and in general, should it be assumed that intersection always goes (is processed) first, before that Union.
2. Does it make sense to generalize that, what is applicable to + and * when in regular arithmetic {i.e (A+B)=(B+A) } Is valid when replacing + by Union and * by Intersection {i.e ( A u B ) = (B u A) or as A*(B+C) is AB +AC then A n ( B u C) → (A n B ) u ( B n C) shoud be ok…)
This is really great for learning the complete possibilities of the Wolfram Language.. Thanks for doing it so thoroughly!. You could paste a picture!.. but you didn’t!. Thanks a lot!
Yes! This is the priority of operations, as noted throughout the lesson. First parentesis, then negation, then intersection, then union. Like the infamous PEDMAS, but for set operations.
Yes, the generalization applies. Imagine it as equivalent operations but operation on different objects.
The logical Or ||, the addition +, the set union, all represent the natural first common operation.
The logical And &&, the multiplication *, the set intersection, all represent the natural second common operation in a ring.
This is a common theme in mathematics, which implies a lot of the basic rules you see, often in linear algebra, are appliable to many many contexts!
To be fair, this is extra knowledge that is only there for your information.
That said, let’s get into it.
The Naive Bayes assumption is that every event is considered independent. This implies there is n probabilities to guess for any set of n events. From there, a Naive Bayes classifier, a machine learning algorithm, will assume every attribute of your dataset is independent. Now, with a lot of data points, noticing where data points differ, you can infer those n probabilities, which allows you afterwards to predict the class based on your guessed probabilities, just multiplying the independent events.
Feel free to check external resources, like this, to get a good understanding of the algorithm. But this is beyond the scope of the course of course, something you would see in an Introduction to AI.
The example in lesson 3 for the probability that the sum of two dice will be even comes up with a probability of 6/11. This seemed odd and a little bit of extra thought suggests that you should not delete duplicate outcomes as they are part of the sample space of outcomes. When you consider duplicate outcomes in both numerator and denominator, you get the more intuitively satisfying answer of a probability of 1/2.
In the material of Lesson 4, when referring to the probability of the event “E” , and the need for Kolmogorov axioms, appears this..
However, k/M must be constant to be relevant, which is problematic<
Every time I “test” the 1/6 of a perfect dice, the number of k ones / M trials is not constant, certainly because random events are not constant. So I don’t believe I get the core idea of why k/M must be constant or why this is “problematic”. From an Statistical perspective that k/M is how we ( or at least I ) estimate (guess) probability. So, I’m confused.
