Questions tagged [probability]

For questions about the teaching of probability, dealing with students misconceptions in probability, and explaining probability theory paradoxes.

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How to prove, without the LOTUS formula, to student that $V[aX+b]= a^2 V[X]$?

The mainstream way to show $V[aX+b]= a^2 V[X]$ is by using LOTUS. However, LOTUS seems to me too powerful and out-of-reach for a last-year high-school student. Therefore I was wondering if we could ...
niobium's user avatar
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-1 votes
1 answer
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Does some textbook on probability use a particular way (described here) of accomplishing the segue into continuous distributions?

Imagine an introductory probability course that assumes the students have had first-year calculus and understand mathematical reasoning. At some point in such a course has explicated several families ...
Michael Hardy's user avatar
5 votes
7 answers
2k views

Should I choose Cox-based Jaynes' approach or Kolmogorov approach to base myself on to teach probabilities to high-school students?

I am planning to become a math high-school teacher and have the following question: What Probability Theory should I base myself on to teach probabilities to students ? The classical approach is via ...
niobium's user avatar
  • 215
7 votes
7 answers
1k views

Special topics for introductory probability

I am helping to design a low-level college course whose purpose is to teach critical thinking, logic, finance and probability. I have been tasked with developing the probability section. I am ...
dt688's user avatar
  • 173
-6 votes
1 answer
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PROBABILITY QUESTION [closed]

How to solve this probability question?f a, b, and c are in the range [0, 1], what is the probability that the quadratic equation ax^2 + bx + c has real solutions? justify your answer! This question ...
Humberto José Bortolossi's user avatar
1 vote
2 answers
142 views

Online Probability Simulation for Compound Events

I'm teaching Grade 9 Probability. We need to compare Experiment Probabilities (Relative Frequencies) with Theoretical Probabilities for Simple and Compound Events. I want to come up with and Online ...
pdmclean's user avatar
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0 votes
1 answer
69 views

Seeking References on Deterministic and Stochastic Phenomena Suitable for High School Students

Can anyone recommend good and didactic references that delve into the dualism between deterministic and stochastic phenomena? Ideally, I'm seeking materials that provide a conceptual explanation along ...
Humberto José Bortolossi's user avatar
1 vote
1 answer
64 views

Discrete Probability Modeling with Desmos or Spreadsheets

In my Finite Math course* almost every section includes a part where students have to create a file (from scratch) in Desmos or in Google Sheets. For example, they use Desmos to plot piecewise linear ...
Aeryk's user avatar
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5 votes
1 answer
1k views

Law of large numbers as a middle school topic?

My daughter (a biologist) is presently teaching also math at a middle school (9th grade, so about 14 years olds). Now the topic in probability seems to be the law of large numbers! More and more I ...
kjetil b halvorsen's user avatar
3 votes
2 answers
353 views

Common mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
Botnakov N.'s user avatar
5 votes
2 answers
324 views

How to teach the concept of probability distribution?

I observed that my students do not understand what a probability distribution is. We do not treat probability axiomatically on the course, so the required level of understanding is knowing all the ...
Tommi's user avatar
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5 votes
1 answer
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Examples of random variables that are measurable with respect to a strict subset of $\mathcal{P}(\Omega)$

When teaching random variables for the first time, most of us say that it is a function $$X : \Omega \to \mathbb{R}$$ without any further restriction. Of course, a more general definition is to say ...
user700974's user avatar
-3 votes
2 answers
152 views

Best natural language(s) for conveying, conceptualizing, teaching, understanding, and learning Probabilistic & Statistical concepts & theory?

English can be precise but it is rather 'flowery' and easily gets in its' own way. East-Asian natural languages like Mandarin, Cantonese, Korean, and Japanese are notorious for permitting the ...
NoYouNaiveBaye's user avatar
-4 votes
1 answer
165 views

Is it weird for an undergrad or grad quant/applied maths(/even pure maths) programme to not teach that probabilities of 0 or 1 will never change? [closed]

Edit: i didn't mean it like this programme should do this or that. i mean other people are accusing me like 'your programme should've had this or that' (actually they're saying that i should know this ...
BCLC's user avatar
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1 answer
525 views

Best books for mathematical statistics self-study?

I'm hoping to start a masters in mathematics in the fall, and am hoping to find a good book on mathematical statistics to study so that I'll be able to take graduate level mathematical statistics once ...
blakedylanmusic's user avatar
0 votes
0 answers
77 views

Code for Urn (containing colored balls) generator? [duplicate]

I am interested in code that generates an urn with various colored balls for an economics experiment. I came across this thread: Urn (containing colored balls) generator? However, the server that the ...
khkellis's user avatar
4 votes
3 answers
283 views

Probability — analytical results instead of simulations

After students learn how to use probabilistic simulations, what strategies can one use to encourage them to understand analytical results anyway? For example, I'm struggling to find a compelling ...
Paula's user avatar
  • 251
1 vote
2 answers
129 views

How can I effectively learn and master Math and Statistics for Data Science?

I completed a BSc in Computer Science recently and am going on to do an MSc in Data Science. However, the only focussed math module I had during CS was in the first year and I didn't do too well. I ...
curiousCoder's user avatar
3 votes
1 answer
189 views

Not sure what a student is misunderstanding on this Stat question

Here is the solution I presented: (a) P(G|B) = (5/24) * (.125) = .026. Since you select ONE senior manager, then P(B) is 4/23. Now, if we are dealing with a second employee who is also a senior ...
Wasp's user avatar
  • 187
6 votes
2 answers
219 views

Difficulty in explaining sample space

I am running into problems explaining to my students in high school what exactly is sample space in probabilities, especially with identical objects. For example, according to Q6.2.3 of this UIC ...
user15538's user avatar
2 votes
1 answer
241 views

What story and one-digit Natural Numbers best fit Bayes' Theorem chart?

Some students have sniveled that most examples of Bayes' Theorem use non-integer numbers. I want to try a Bayes' Theorem chart that uses just single digit Natural Numbers $\le 9$. To complete the ...
user avatar
50 votes
21 answers
16k views

How to explain that winning the lottery is not a 50/50 distribution?

When casually discussing with my 13 yo child about probabilities, he told me there is a 50% chance to win at the lottery To what I said no, there is a 1 chance over 90 million (I roughly estimated ...
WoJ's user avatar
  • 1,388
1 vote
2 answers
222 views

Cautionary tales for Gaussians/the central limit theorem?

In any undergraduate course on probability, one of the fundamental results that is discussed is the central limit theorem. Apart from the interesting mathematics needed to prove the result (most often ...
user78270's user avatar
  • 111
4 votes
0 answers
108 views

Transition into a non-explicit sample space world

This is one of the problem taunting me over years while explaining probability. In most of the high-school as well as graduate textbooks, there are at most very few lines to deal with this problem. ...
hanugm's user avatar
  • 259
3 votes
4 answers
502 views

Best resources for probability/statistics textbooks

I'm looking for a good textbook introduction to probability/statistics that a first/second year undergrad math student could use! I'd like a book that emphasizes theory over procedure. I'd prefer an ...
Lex_i's user avatar
  • 496
4 votes
1 answer
100 views

EdX Courses for Self-Study

I have been independently considering two edX courses in mathematics. The first, a course on probability theory drawing from a financial crisis case study, appeared to me plausibly comparable in ...
user13350's user avatar
2 votes
1 answer
133 views

How to emulate erf and/or the Normal Inverse function in Moodle?

I've just discovered "calculated questions" in Moodle and I'm trying to create a simple one where I would be asking the student to find the probability that an observation from a standard normal ...
Arnaud Mortier's user avatar
12 votes
9 answers
1k views

Why do you need to distinguish between apparently identical objects in probability?

In the math class I taught today I was asked a question, and I was unable to give a good answer. The problem was as follows: A certain factory produces throat tablets. In each pack, there are from 48 ...
Arthur's user avatar
  • 401
4 votes
1 answer
342 views

The most transparent exposition of Bayes' Theorem

I am seeking the most transparent exposition of Bayes' Theorem (for undergraduates). I would prefer to avoid mentioning "prior" and "posterior," and instead focus on frequencies. The Wikipedia entry ...
Joseph O'Rourke's user avatar
4 votes
2 answers
101 views

A Markov chain demonstration that doesn't require computers

I have a large probability class and would like to do some memorable demonstrations of Markov chains for them. Does anyone have any recommendations for a "low-tech" demo that doesn't involve ...
Tom Solberg's user avatar
5 votes
2 answers
168 views

Better ways to explain mutually exclusiveness and dependency of events

I am teaching probability on mutually exclusiveness and dependency of events. Let me take a simple example as follows. A box contains 2 red balls and 3 purple balls. They are identical except for ...
Display Name's user avatar
2 votes
1 answer
184 views

What does "Four selected students are not born in the same months" mean?

I am teaching on probability. I found a question that seems to be ambiguous as follows. Four students are randomly chosen from a place. Assuming the birthdays of people are equally likely to occur ...
Display Name's user avatar
4 votes
1 answer
280 views

How to explain the sample space of Monty Hall problem?

I am now pretending to be a newbie student. I write the following sample space for the Monty Hall problem (It is a famous brain teaser, I assume you know it). $$ S=\{ (C,G1),(C,G2), (G1,G2), (G2,G1) \...
Display Name's user avatar
6 votes
1 answer
135 views

Why emphasize moment generating function over characteristic function in a probability course?

I've noticed that some undergraduate introductory probability textbooks and courses emphasize or seem to prefer the moment generating function $m(t) = \mathbf E(e^{tX})$ of a random variable $X$ ...
eternalGoldenBraid's user avatar
5 votes
1 answer
202 views

Encouraging students to learn probability

Background: I’m not exactly a Math educator, but I’m currently a TA of an elementary algebra course aiming at students of age 14-15. I found that a lot of people have misconception about probability ...
tonychow0929's user avatar
0 votes
0 answers
157 views

What is the best term for "probability measure" in an undergrad introduction to probability course?

The function $P$ that takes an event $A$ as input and returns the probability $P(A)$ as output is called a "probability measure" when we are developing probability using measure theory. I have also ...
eternalGoldenBraid's user avatar
3 votes
2 answers
271 views

Urn (containing colored balls) generator?

I am looking for a nice app that would enable me to create "automatically" nice urns filled with balls of two different colors, following the illustration below (actual colors do not matter, being ...
L. M.'s user avatar
  • 133
2 votes
1 answer
166 views

Why aren't Bayesian Networks and Variable Elimination introduced earlier?

Throughout my undergrad, I dreaded probability. I hated it, I was horrible in it, I just never got it, and felt stupid when the professors used "summation/marginalization" equations out of the blue to ...
user3508551's user avatar
7 votes
6 answers
630 views

Neat topics or problems to include in a probability class

I'd like to get suggestions for neat topics or problems to include in an undergraduate, upper division Introduction to Probability class. Many people have taught probability for many years and I'm ...
eternalGoldenBraid's user avatar
4 votes
3 answers
318 views

How to convince the following probability problem to highschool student?

I will present two problems alongside solutions, student is doing problems of type I like a cakewalk but has several issues with the problems of type 2; Type I Consider an experiment of rolling ...
hanugm's user avatar
  • 259
12 votes
4 answers
2k views

Real-world Markov chains

I will give a talk to undergrad students about Markov chains. I would like to present several concrete real-world examples. However, I am not good with coming up with them beyond drunk man taking ...
Marcel's user avatar
  • 222
5 votes
2 answers
102 views

Examples showing necessity and advantage of emperical definition of probability

As the title, I want to ask for some examples showing necessity and advantage of emperical definition of probability in teaching high school students. We knew that the classical definition has some ...
Tien Kha Pham's user avatar
3 votes
8 answers
723 views

How to best explain that the sum of conditional probabilities still sum to 1

I am comfortable with explaining to my high school students that for an event $A$ we have that $P(A) + P(A^C) = 1$ But what is the best way to help students realize that $P(A \mid B) + P(A^C \mid ...
Improve's user avatar
  • 1,881
19 votes
5 answers
2k views

Common misconceptions in high school probability curriculum

I am teaching probability to high school students. The material we are covering is pretty standard and includes: Introducing how to calculate the probability of events, e.g. coin flips, card draws, ...
Improve's user avatar
  • 1,881
12 votes
5 answers
955 views

Moving from discrete probability distributions to continuous ones

I'm teaching an introductory statistics class at a community college, and we've just finished a unit on discrete probability. At the moment, the students' conception of the probability of an event A ...
Jared's user avatar
  • 2,223
8 votes
3 answers
260 views

Mathematical Task with Various Solutions

I need to come up with a mathematical task for middle school (9th grade), which involves either algebra, functions, probability or statistics (anything but geometry actually). My problem is, that the ...
user2899944's user avatar
7 votes
0 answers
96 views

Literature on teaching and learning probability

In an earlier question, the book Exploring Probability in School (2005) (Link) was mentioned. It gives an overview of the research on the teaching and learning of probability up to that point. Does ...
Dag Oskar Madsen's user avatar
4 votes
3 answers
317 views

Monty Hall challenge

Thinking about the counterintuitive Monty Hall Problem (stick or switch?), revisited in this ME question, I thought I would issue a challenge: Give in one (perhaps long) sentence a convincing ...
Joseph O'Rourke's user avatar
4 votes
3 answers
134 views

Example illustrating use of additivity of (probability) measures before introducing independence and conditioning

I would like to be able to illustrate why additivity is a natural property to assume for probability measures. While it is relatively simple to give a short hand waving intuitive analogy to volume or ...
tilo.wiklund's user avatar
4 votes
2 answers
154 views

suggestion for probability and stats text to suit my learning style.

I am of a mature aged learner and want to learn probability and statistics at a senior high school (k12) and undergraduate level for my own general understanding. I have discovered in the past that I ...
hgtest35's user avatar