Seeing Theory: A Visual Introduction to Probability and Statistics(seeing-theory.brown.edu)
seeing-theory.brown.edu
Seeing Theory: A Visual Introduction to Probability and Statistics
https://seeing-theory.brown.edu/index.html
93 comments
On the topic of the Monty Hall problem, what helped me "believe" it more was if you change it to 1,000,000 doors, still with only 1 car, and the rest goats. You choose 1 door. The host then opens up 999,998 other doors, which all contain goats. So there are 2 doors left. Your door, and the only other door the host didn't open. Do you feel at a gut level that you should switch?
I see this argument a lot and for some reason it doesn't help me with the intuition at all. If you (wrongly) get caught up on the fact that the remaining door and your pick have the same initial probabilities of being a car, then you'll still think that switching doesn't make a difference even in the million-door case.
Here's what works for me:
- the switching strategy always gives you the opposite of your initial choice
- the initial probabilities are 2/3 goat and 1/3 car so by switching you get 2/3 car and 1/3 goat
Here's what works for me:
- the switching strategy always gives you the opposite of your initial choice
- the initial probabilities are 2/3 goat and 1/3 car so by switching you get 2/3 car and 1/3 goat
When Monty opens doors he uses 2 pieces of information: the door you picked and the correct door.
After he opens 999,998 doors he has given you quite a bit of information. There is a 1/1000000 chance though that he has given you no information (you picked the correct door)
But you're right that thinking about it in partitions also makes sense. You try to pick a partition size 1 that contains the prize, while Monty picks the partition size 999,999, if you agree with his partition and it has the prize you get it
After he opens 999,998 doors he has given you quite a bit of information. There is a 1/1000000 chance though that he has given you no information (you picked the correct door)
But you're right that thinking about it in partitions also makes sense. You try to pick a partition size 1 that contains the prize, while Monty picks the partition size 999,999, if you agree with his partition and it has the prize you get it
I think the point a lot of people miss is that the trick to understanding the 3-door question and the 1000000-door question is the same trick. If you don't grok the trick, the 1000000-door example might make it easier to grok, but there's a fair chance it won't.
To muddy the waters further, it's not always understood that in the 1000000-door case, 999998 other doors are opened (as evidenced by discussion elsewhere in these threads). Sometimes people think it's still just one door. I suspect this is because the original problem is usually stated as "...Monty Hall then opens one of the doors you didn't pick" and because people suggesting the 1000000-door often just say "...what if there were one million doors?"
To muddy the waters further, it's not always understood that in the 1000000-door case, 999998 other doors are opened (as evidenced by discussion elsewhere in these threads). Sometimes people think it's still just one door. I suspect this is because the original problem is usually stated as "...Monty Hall then opens one of the doors you didn't pick" and because people suggesting the 1000000-door often just say "...what if there were one million doors?"
Rather than going through this convoluted kinda similar problem, I find it easier to stick to the original one.
Get a piece of paper. Draw all possible outcomes, 9 total. ( Car is behind door 1 you pick 1, Car is behind door 1 you pick 2...). 3 of the 9 result in success.
Now draw the outcomes again but switch every time. 6 out of 9 outcomes are a success.
Get a piece of paper. Draw all possible outcomes, 9 total. ( Car is behind door 1 you pick 1, Car is behind door 1 you pick 2...). 3 of the 9 result in success.
Now draw the outcomes again but switch every time. 6 out of 9 outcomes are a success.
What gave me the intuition for it was to realise that Monty is giving you the choice to either stick with your original door, or take the sum of the prizes between the other two doors.
(He also opens one of those two doors to reveal a goat, but you already knew that one of them had a goat so that doesn't give you any additional information.)
(He also opens one of those two doors to reveal a goat, but you already knew that one of them had a goat so that doesn't give you any additional information.)
Many people have suggested this "intuitive" explanation. But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door.
"But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door."
It SHOULD be clear, because you have two givens: 1) Monty never reveals the car. 2) He opens all the doors except 1.
It SHOULD be clear, because you have two givens: 1) Monty never reveals the car. 2) He opens all the doors except 1.
"2) He opens all the doors except 1"
How is this a given exactly? In the original problem he only opens 1 other door. Now that also happens to be all doors except 1, but from just the 3 door problem that seems more coincidental than a fundamental part to the question
How is this a given exactly? In the original problem he only opens 1 other door. Now that also happens to be all doors except 1, but from just the 3 door problem that seems more coincidental than a fundamental part to the question
It's a given by the person who mentioned 999,998 doors. I think you're missing the point, but I won't pursue this further.
Sorry, but you are the one missing the point. The person who mention 999,998 doors didn't give any reasoning for why that would be the logical extension of the problem.
Obviously, you and I know it is, but the person grappling with the Monty Hall problem is right in not being convinced of that just because someone says it is!
Obviously, you and I know it is, but the person grappling with the Monty Hall problem is right in not being convinced of that just because someone says it is!
The rationale for opening 999,998 doors in my example versus the 1 door left, is that in both examples, Monty Hall opens every door except the one you're on, and 1 other door. It happens that in the normal Monty Hall, if you open every door except the one you're on, and 1 other door, you have only opened one door.
Monty Hall is asking you a simple question, whether or not you should switch, and so in my example of 1,000,000 whether or not you open 999,998 doors, or 1 door, you will always have worse odds to win if you don't switch to another door. Removing 999,998 doors just takes the proposition to an extreme.
Another component to utilize one's intuition using the 999,998 example, would be to imagine the game being played 3 times in a row. What are the odds that not switching will help you? So basically, not switching is disregarding everything Monty Hall is doing. You are either behind a door or you are not. You don't switch. If that is how you play the game, your chance of choosing right when not switching is 1/1,000,000 each game, or 1/10^18 for it to happen 3 times in a row. Now, consider what Monty is doing. He's removing every chair but two, yours and another. If the odds of you winning are 1/1,000,000 if you don't switch, What are the odds of doing _the opposite_? Since there are only two options, the probability of winning if you switch is 1-1/1,000,000, or 999,999/1,000,000, as the sum of the probabilities of all possible events has to add up to 1.
The "999,998" chairs removed example is an attempt at making the dichotomy between "stay" and "switch" more extreme, so that you would feel it in your gut rather than trying to mentally account for the moving pieces.
I'm always interested in improving my ability to explain these kinds of phenomena, and I appreciate the pointing out of why the dots don't get connected for some with the example.
Monty Hall is asking you a simple question, whether or not you should switch, and so in my example of 1,000,000 whether or not you open 999,998 doors, or 1 door, you will always have worse odds to win if you don't switch to another door. Removing 999,998 doors just takes the proposition to an extreme.
Another component to utilize one's intuition using the 999,998 example, would be to imagine the game being played 3 times in a row. What are the odds that not switching will help you? So basically, not switching is disregarding everything Monty Hall is doing. You are either behind a door or you are not. You don't switch. If that is how you play the game, your chance of choosing right when not switching is 1/1,000,000 each game, or 1/10^18 for it to happen 3 times in a row. Now, consider what Monty is doing. He's removing every chair but two, yours and another. If the odds of you winning are 1/1,000,000 if you don't switch, What are the odds of doing _the opposite_? Since there are only two options, the probability of winning if you switch is 1-1/1,000,000, or 999,999/1,000,000, as the sum of the probabilities of all possible events has to add up to 1.
The "999,998" chairs removed example is an attempt at making the dichotomy between "stay" and "switch" more extreme, so that you would feel it in your gut rather than trying to mentally account for the moving pieces.
I'm always interested in improving my ability to explain these kinds of phenomena, and I appreciate the pointing out of why the dots don't get connected for some with the example.
> The rationale for opening 999,998 doors in my example versus the 1 door left, is that in both examples, Monty Hall opens every door except the one you're on, and 1 other door. It happens that in the normal Monty Hall, if you open every door except the one you're on, and 1 other door, you have only opened one door.
And the rationale for opening 1 other door in the million door example is that in both examples the host is opening 1 other door. The normal Monty Hall problem is usually formulated such that the host opens 1 other door, not that he opens all other doors. As you noted, the two formulations are equivalent in with 3 doors, but with more than 3 doors, they're not. I just don't see why it's "intuitive" that if the number of doors is increased, the natural extension of the game is that the host opens all other doors that don't have the prize. In fact I'd argue the opposite.
Imagine an actual Monty Hall game with 4 doors. The contestant opens 1 door with a goat, and the host might open (a) 1 other door with a goat or (b) 2 other doors with a goat. Both are valid, reasonable, but different extensions of the game. In both versions, the best strategy for the contestant is to switch[1], because in both versions the host is giving her extra information. But in version (b) he's giving her much more information than in version (a). Of course it's much easier to intuit in version (b) that switching is better, but it's not clear to me why version (b) rather than (a) is the natural 4-door analog to the 3-door Monte Hall game. If a 4 door version were played in real life, it's far more likely IMO that version (a) would be played. In (a) the host gives a little bit of extra info where the prize is, without giving away the solution. And in this case you need a much better model to see why this is the case in stead of relying on intuition and analogy.
[1] A usually unstated assumption in most formulations is that the host must open another door with a goat if the contestant initially chooses a wrong door. In an actual TV show the host will likely have discretion whether he opens another door at all, to increase suspension and not become predictable in repeated games. In this case the problem becomes much more difficult as you need to model the strategy of the host. All of thise and pretty much any other solutions and explanations on pretty much every forum is already extensively documented in Wikipedia (https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...).
And the rationale for opening 1 other door in the million door example is that in both examples the host is opening 1 other door. The normal Monty Hall problem is usually formulated such that the host opens 1 other door, not that he opens all other doors. As you noted, the two formulations are equivalent in with 3 doors, but with more than 3 doors, they're not. I just don't see why it's "intuitive" that if the number of doors is increased, the natural extension of the game is that the host opens all other doors that don't have the prize. In fact I'd argue the opposite.
Imagine an actual Monty Hall game with 4 doors. The contestant opens 1 door with a goat, and the host might open (a) 1 other door with a goat or (b) 2 other doors with a goat. Both are valid, reasonable, but different extensions of the game. In both versions, the best strategy for the contestant is to switch[1], because in both versions the host is giving her extra information. But in version (b) he's giving her much more information than in version (a). Of course it's much easier to intuit in version (b) that switching is better, but it's not clear to me why version (b) rather than (a) is the natural 4-door analog to the 3-door Monte Hall game. If a 4 door version were played in real life, it's far more likely IMO that version (a) would be played. In (a) the host gives a little bit of extra info where the prize is, without giving away the solution. And in this case you need a much better model to see why this is the case in stead of relying on intuition and analogy.
[1] A usually unstated assumption in most formulations is that the host must open another door with a goat if the contestant initially chooses a wrong door. In an actual TV show the host will likely have discretion whether he opens another door at all, to increase suspension and not become predictable in repeated games. In this case the problem becomes much more difficult as you need to model the strategy of the host. All of thise and pretty much any other solutions and explanations on pretty much every forum is already extensively documented in Wikipedia (https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...).
pyhtel, I gave it a go at explaining the rationale in this comment I made below: https://news.ycombinator.com/item?id=24643272
em500, I gave it a go at explaining the rationale in this comment I made below: https://news.ycombinator.com/item?id=24643272
But this raises a different problem with intuition:
If Monty doesn't know where the car is, then if 999,998 doors were opened showing goats, leaving two doors, the odds that the car is behind your door or behind the remaining door is 1:1 ... this defies many people's intuition.
The difference between the two cases is that, if Monty knows where the car is, then his opening 999,998 doors with goats behind them is exactly what we expect, whereas if he doesn't know where the car is, then his opening 999,998 doors with goats behind them is an extraordinarily unlikely event. But if that does happen despite being extraordinarily unlikely, then there's still a 50% chance that the car is behind your door.
If Monty doesn't know where the car is, then if 999,998 doors were opened showing goats, leaving two doors, the odds that the car is behind your door or behind the remaining door is 1:1 ... this defies many people's intuition.
The difference between the two cases is that, if Monty knows where the car is, then his opening 999,998 doors with goats behind them is exactly what we expect, whereas if he doesn't know where the car is, then his opening 999,998 doors with goats behind them is an extraordinarily unlikely event. But if that does happen despite being extraordinarily unlikely, then there's still a 50% chance that the car is behind your door.
My intuition is that:
1) I will probably lose when Monty opens a car door. 2) If I don't, I am really gambling between whether I made a 1-in-a-million pick or Monty did (in the choice of which door to leave shut), which obviously has even odds.
Interestingly, by compressing this problem back down to the 3-door version, it makes it pretty obvious why that's the case (and aligns with people's intuition about the original problem). Also interesting that in this case, even if the intuition is wrong (that 'obviously' they must have picked the car), the outcome (sticking with the chosen door) is an optimal strategy.
1) I will probably lose when Monty opens a car door. 2) If I don't, I am really gambling between whether I made a 1-in-a-million pick or Monty did (in the choice of which door to leave shut), which obviously has even odds.
Interestingly, by compressing this problem back down to the 3-door version, it makes it pretty obvious why that's the case (and aligns with people's intuition about the original problem). Also interesting that in this case, even if the intuition is wrong (that 'obviously' they must have picked the car), the outcome (sticking with the chosen door) is an optimal strategy.
What's unintuitive about the Monty Hall problem is the difference between mathematical Monty and a psychological Monty. It is easy to imagine Monty almost only opening doors in case the guest choose the car, and not opening anything in case they choose a goat. So, when presented with the choice, a cautious guest will hesitate to change. If, however, the guest knows from previous shows that Monty will ALWAYS open a goat door, it is still mentally hard to change the cautious strategy.
The mathematics behind probability and statistics is about as ripe for intuition as calculus and linear algebra. A lot of it really comes down to counting in probability (calculus/measure theory for the continuous case) and quantifying properties about probability distributions for statistics.
The really hard part is the modelling part, where you transform the problem to a mathematical statement and vice versa. It's very easy to misinterpret both the problem in terms of mathematics and the mathematical result in terms of the problem. All the wrong answers to brain teasers like the monty hall problem, the tuesday boy problem etc., are right answers to the wrong question.
Unfortunately, in education we do not seem to want to discuss the modelling part on equal terms with the theory. We seem to be okay with solving the entire problem, or solving just the theoretical part with no regards to the application, but expressing just the mathematical problem to be solved is never appreciated. In a calculus setting, this could be deriving the answer to some physical problem depends on the solution of some partial differential equation -- even if you do not have the tools to solve it outright.
My guess is that it's just easier to teach theory with clear cut answers. Modelling the real world is ambiguous and hard.
The really hard part is the modelling part, where you transform the problem to a mathematical statement and vice versa. It's very easy to misinterpret both the problem in terms of mathematics and the mathematical result in terms of the problem. All the wrong answers to brain teasers like the monty hall problem, the tuesday boy problem etc., are right answers to the wrong question.
Unfortunately, in education we do not seem to want to discuss the modelling part on equal terms with the theory. We seem to be okay with solving the entire problem, or solving just the theoretical part with no regards to the application, but expressing just the mathematical problem to be solved is never appreciated. In a calculus setting, this could be deriving the answer to some physical problem depends on the solution of some partial differential equation -- even if you do not have the tools to solve it outright.
My guess is that it's just easier to teach theory with clear cut answers. Modelling the real world is ambiguous and hard.
It's because math courses are crazy expensive and teaching both modelling and theory together would mean taking way longer (read costing way more) or making the failure rate (cost) way higher.
Wow, strong disagree. Once you develop intuition, probability is really quite intuitive. This kind of course should be working to develop this intuition — like the conditional probability examples and the CLT examples. The computational examples inline really help here.
The Monte Hall problem is more of a curiosity than a fundamental principle!
(Was a TA in undergrad engineering probability for 2 years, saw my share of learners.)
The Monte Hall problem is more of a curiosity than a fundamental principle!
(Was a TA in undergrad engineering probability for 2 years, saw my share of learners.)
I can't argue with "once you develop intuition, probability is intuitive". I was arguing that lessons starting with E(X)=... basically stop the majority of people from getting to the point, where they see how their "initial intuition" is wrong.
Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class.
Monte Hall was one example. The birthday problem and the base rate fallacy are two more [1][2]. The result seems obvious but most people get these wrong.
With a couple of papers or books by Kahneman and Tversky in hand we can generate an almost infinite list of simple statistics/probability questions, which most people get wrong. Let people make some mistakes, before dumping the theory on them.
[1]https://en.wikipedia.org/wiki/Base_rate_fallacy [2]https://en.wikipedia.org/wiki/Birthday_problem
Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class.
Monte Hall was one example. The birthday problem and the base rate fallacy are two more [1][2]. The result seems obvious but most people get these wrong.
With a couple of papers or books by Kahneman and Tversky in hand we can generate an almost infinite list of simple statistics/probability questions, which most people get wrong. Let people make some mistakes, before dumping the theory on them.
[1]https://en.wikipedia.org/wiki/Base_rate_fallacy [2]https://en.wikipedia.org/wiki/Birthday_problem
> Monte Hall was one example.
Monty Hall is not a good example, unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat. Just look at the discussions in the comment here.
Monty Hall is not a good example, unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat. Just look at the discussions in the comment here.
> Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class.
Again, strong disagree. Probability has been understood at a quantitative level since Laplace (1812). Modern measure-theoretic probability dates from Kolmogorov's foundational work (1933). All these years later, we really know this stuff.
Specifically: A lot of general-purpose, powerful tools have been developed. Distribution theory, the strong law of large numbers, the CLT, maximum likelihood, L2 theory for estimation.
Depending on your goals, these or related tools are capable of addressing a wide range of problems. The priority of the first few courses should be to impart mastery of a selection of these general-purpose tools, so that students know how to analyze problems probabilistically. This is where intuition comes from.
Gotcha problems like Monte Hall are not getting you to this goal!
One could argue that MHP can motivate the notion of conditioning, but I think fundamentally the MHP is verbal legerdemain. That is, you state the problem such that the conditioning is implicit in the actions, and people don't notice it. Recall that the questioner obtains "victory" when, after presenting the problem, the answerer is confused and gives the wrong answer. I don't like that approach as a teaching tool.
I'm also skeptical of the Birthday Problem and the Kahneman-Tversky surprises. I see value in these surprising conundrums (the Birthday Problem is in volume 1 of Feller, so it has a pedigree) only to the extent that they motivate the utility of general-purpose analytic tools. They are an appetizer, not the main dish.
Again, strong disagree. Probability has been understood at a quantitative level since Laplace (1812). Modern measure-theoretic probability dates from Kolmogorov's foundational work (1933). All these years later, we really know this stuff.
Specifically: A lot of general-purpose, powerful tools have been developed. Distribution theory, the strong law of large numbers, the CLT, maximum likelihood, L2 theory for estimation.
Depending on your goals, these or related tools are capable of addressing a wide range of problems. The priority of the first few courses should be to impart mastery of a selection of these general-purpose tools, so that students know how to analyze problems probabilistically. This is where intuition comes from.
Gotcha problems like Monte Hall are not getting you to this goal!
One could argue that MHP can motivate the notion of conditioning, but I think fundamentally the MHP is verbal legerdemain. That is, you state the problem such that the conditioning is implicit in the actions, and people don't notice it. Recall that the questioner obtains "victory" when, after presenting the problem, the answerer is confused and gives the wrong answer. I don't like that approach as a teaching tool.
I'm also skeptical of the Birthday Problem and the Kahneman-Tversky surprises. I see value in these surprising conundrums (the Birthday Problem is in volume 1 of Feller, so it has a pedigree) only to the extent that they motivate the utility of general-purpose analytic tools. They are an appetizer, not the main dish.
> Probability has been understood at a quantitative level since Laplace [...] and Kolmogorov.
Which indicates it is roughly as hard as partial differential equations, the theory of relativity and just a tiny bit easier than some of the quantum mechanics.
This is pretty unintuitive for a subject, which mostly relies on multiplication and addition.
The dozen or so posts discussing the intuition of the Monty Hall Problem are a case in point.
> They are an appetizer, not the main dish.
This is certainly true.
Which indicates it is roughly as hard as partial differential equations, the theory of relativity and just a tiny bit easier than some of the quantum mechanics.
This is pretty unintuitive for a subject, which mostly relies on multiplication and addition.
The dozen or so posts discussing the intuition of the Monty Hall Problem are a case in point.
> They are an appetizer, not the main dish.
This is certainly true.
"Once you develop intuition, probability is really quite intuitive"
That's a tautology.
Plenty of studies, such as the work by Kahneman and Tversky, show that humans by default have incorrect statistical intuitions. These faulty intuitions are hard to overcome, even by a considerable amount of training.
> The Monte Hall problem is more of a curiosity than a fundamental principle!
It's quite straightforward conditional probability. That so many people, including trained mathematicians, get it wrong is quite illustrative. And it's not unique ... the coins and drawers problem is similar, and one can craft many others. MH is not a mere curiosity, it's simply well known.
That's a tautology.
Plenty of studies, such as the work by Kahneman and Tversky, show that humans by default have incorrect statistical intuitions. These faulty intuitions are hard to overcome, even by a considerable amount of training.
> The Monte Hall problem is more of a curiosity than a fundamental principle!
It's quite straightforward conditional probability. That so many people, including trained mathematicians, get it wrong is quite illustrative. And it's not unique ... the coins and drawers problem is similar, and one can craft many others. MH is not a mere curiosity, it's simply well known.
> It's quite straightforward ...
No it is not, unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat. Just look at the discussions in the comment here.
No it is not, unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat. Just look at the discussions in the comment here.
> unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat.
That has been part of the explicit problem ever since it was first presented back in 1975.
"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice?"
That has been part of the explicit problem ever since it was first presented back in 1975.
"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice?"
It does not clearly say that it is Monty’s procedure to
- allways open a door
- allways open a door with a goat
- and not open a door at random
Often discussion of the solution reveals that this is not clear.
- allways open a door
- allways open a door with a goat
- and not open a door at random
Often discussion of the solution reveals that this is not clear.
There are three doors. You have picked one, leaving two other doors. It absolutely explicitly says he opens one door. The only options are a goat or a car. If it was a car, you would have lost already and so there is no problem. If it was random, you still get the same information (what is behind one of the unpicked doors).
> If it was random, you still get the same information
No, if both you and Monty pick a door at random, there’s 1/3 chance of a car behind each door. If Monty’s door reveals a goat, it’s 50/50 for the remaining two doors. It’s mandatory to specify Monty’s procedure precisely.
No, if both you and Monty pick a door at random, there’s 1/3 chance of a car behind each door. If Monty’s door reveals a goat, it’s 50/50 for the remaining two doors. It’s mandatory to specify Monty’s procedure precisely.
This is goalpost moving that has nothing to do with the original point. If people misunderstand the conditions of the problem, that has nothing to do with intuitions about probability.
I won't respond further.
I won't respond further.
> nothing to do with the original point.
Agree, but my point is that the Monty puzzle is a bad example to use educational if not careful.
Agree, but my point is that the Monty puzzle is a bad example to use educational if not careful.
It is a tautology, but we are studying teaching, so maybe that's not unexpected? ;-)
My point is that the goal of the course should be to understand principles, not to teach people that their existing intuition is faulty. Who cares about their prior condition of ignorance?
For more, see my reply nearby.
My point is that the goal of the course should be to understand principles, not to teach people that their existing intuition is faulty. Who cares about their prior condition of ignorance?
For more, see my reply nearby.
I'm always wondering the answer to this question:
Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get moments of a distribution, we multiply to convolve, we multiply to get the work done on an object. I suppose the answer is multiplication allows us to scale some function f(x) with some function g(x). But I guess I want something deeper and I feel like I'm missing it.
Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get moments of a distribution, we multiply to convolve, we multiply to get the work done on an object. I suppose the answer is multiplication allows us to scale some function f(x) with some function g(x). But I guess I want something deeper and I feel like I'm missing it.
We are very good at finding correlations. It is still very hard to prove causality in natural phenomena from experiments, specially when we cannot control them. This became blatantly obvious in the covid outbreak where nobody had a clue for months about whether masks would help or not. Edit to clarify: It is very hard to prove to causality and be sure that you did not mess up.
You are right about people confusing causality and correlation.
Otherwise this site wouldn't be so funny:
https://www.tylervigen.com/spurious-correlations
You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist.
There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.
You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist.
There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.
> But on probability your gut-feel will always fool you.
Yes, this is especially true when first learning; however, one can still develop intuition so that it serves as an invaluable motivator and guide through difficult problems.
Yes, this is especially true when first learning; however, one can still develop intuition so that it serves as an invaluable motivator and guide through difficult problems.
My professor for statistics (he was quite famous in the field) talked about Monty Hall, but made clear that he will not give a solution because of science-political reasons.
I am incredibly curious what he meant by "science-political reasons."
Bringing up Monty Hall at a table full of tech people has always resulted in an argument that will not end until one person gets the rest of us to admit we are wrong and changing doors is the same probability as staying with the same door.
Me too, but he refused to explain. I think there must have been a time where choosing a side was able to end friendships.
Why are you calling the Monty Hall problem for the goat problem?
It is known in academic circles as Monty hall and when it pops up in popular media, it is also referred to as Monty hall.
It is known in academic circles as Monty hall and when it pops up in popular media, it is also referred to as Monty hall.
Thanks for pointing this out. While Google and Wikipedia confidently redirect me to the Monty Hall article, which does mention a goat at some point, the common name for it in English is "Monty Hall Problem".
In other languages it's a three-door-problem or the goat problem, but given that it's such a good example for so many things in psychology and maths, I should be using the most common name in each language.
From Wikipedia[0]
... It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990a):
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
[0] https://en.wikipedia.org/wiki/Monty_Hall_problem
... It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990a):
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
[0] https://en.wikipedia.org/wiki/Monty_Hall_problem
"The law of Small Numbers" chapter 10?
A couple of years ago I was just learning Python and was playing around with matplotlib. Running simulation of a dice roll 100, 1000, 10,000, 100,000, and 1,000,000 times started to show how the distribution starts to catch up with the expected 1/6th probability of each face. I was thinking how good it would be to teach young students this way.
Definitely! Also, not just young students. If you can get over code-phobia, doing random experiments in a class can be really illustrative. When I teach hypothesis testing, I always teach it both from a simulation perspective and from a traditional perspective.
For one, by doing the simulation part directly it's easier to see the "under repeated sampling..." logic inherent in frequentist procedures. Additionally, it's possible to do simulation-based procedures where traditional methods break down (think: permutation tests).
For one, by doing the simulation part directly it's easier to see the "under repeated sampling..." logic inherent in frequentist procedures. Additionally, it's possible to do simulation-based procedures where traditional methods break down (think: permutation tests).
In an effort to reduce screen time, I recently tried to instigate a game of classic table-top Dungeons & Dragons. And I swear, kids were even more interested in the BigInt N-sided die function I cribbed in a python shell than any demons or demigods ;)
Seeing Theory interactivity is very interesting. I think if there is one canonical example to tie it all together it would be something akin to "estimate the likelihood of an extremely rare event". Say, you're a top astrophysicist at NASA and you have to give the President a briefing on the improbability not impossibility of an extinction level asteroid event. And you must justify how those beliefs are informed by and change with data. It ties everything together: physically based world models, event spaces, conditional probabilities, monte carlo sampling and entropy estimation. And would be really fun to boot!
Seeing Theory interactivity is very interesting. I think if there is one canonical example to tie it all together it would be something akin to "estimate the likelihood of an extremely rare event". Say, you're a top astrophysicist at NASA and you have to give the President a briefing on the improbability not impossibility of an extinction level asteroid event. And you must justify how those beliefs are informed by and change with data. It ties everything together: physically based world models, event spaces, conditional probabilities, monte carlo sampling and entropy estimation. And would be really fun to boot!
I had a similar idea to teach physics, well mostly scientific thinking and process by using a physics game engine for our world or some esoteric one and then asking them to perform some actions by creating theories of how the system behaves. Moving from a qualitative analysis of the system to building concrete quantitative theories and then in later stages having their simple theories fail as we deal with more complex interactions with the system and having them adapt their models of this world or rethink another.
Imagine how much their thought process would change if they intimately understood how scientific modelling works.
Imagine how much their thought process would change if they intimately understood how scientific modelling works.
A couple of years ago I was also a great fan of this paradigm where you try to convince yourself that you understand a math concept by coding/simulating it (a procedural, rather than conceptual understanding, if you will). Here for instance I studied the so-called "Secretary Problem", using the tools you mention:
http://cjauvin.blogspot.com/2012/12/find-true-love-on-dating...
http://cjauvin.blogspot.com/2012/12/find-true-love-on-dating...
You can get quite deep this way.
Generative models map well to programming concepts. Mixtures are quite similar to composition, and hierarchical models can be understood as inheritance. Lots of classical models like HMM, LDA, etc are quite similar to those presented in the GoF book in the sense they combine composition and inheritance in some particularly interesting manner.
Generative models map well to programming concepts. Mixtures are quite similar to composition, and hierarchical models can be understood as inheritance. Lots of classical models like HMM, LDA, etc are quite similar to those presented in the GoF book in the sense they combine composition and inheritance in some particularly interesting manner.
On a different thread this morning someone bemoaned the lack of statistical education - a sentiment that is widespread among people who have studied and worked with statistics and probability. It is really exciting to see pedagogical tools that help explain basic but important concepts like distributions and sampling. Great work.
Agreed, this is extremely well-done. Even worse than the general lack of statistical education, I feel the teaching of statistics and probability suffers of the same problems as calculus/real analysis. Introductory statistics classes ramble at length about how random variables are functions from a probability space to a measurable space, but everyone who actually 'gets' the concept behind it eventually thinks in terms of realizations (i.e. much more similarly to what this tutorial does).
Intuition without theory is shallow, but theory without intuition just leads to you eventually forgetting the theory.
I’ll copy my comment from other place in this thread, because I think it might be relevant here. I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis). A combination of intuition AND rigor is hard to come by. With that said...
What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either.
I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs.
I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either.
I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs.
I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
> I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis)
My background is computer science and I had a similar experience. Just a caveat: I'm not arguing that we should stop teaching theory, quite the contrary: most of the times we err on the side of the fluff. In particular, the fact that many reputable institutions are cutting formal logic, computability theory, etc. from their CS curriculums is an absolute disgrace. Intuition is hard to teach (easy to fall into the 'monads are burritos' trap) and it's something you have to work for yourself if you want to develop. My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you.
I honestly don't think it really matters what book you are studying as an introduction to a subject, as usually introductory courses are teaching well-established theory that everyone knows/agrees on. If you have no prior knowledge, a decent starting point is this: https://www.amazon.com/gp/product/1981369198/ the author's website has similar content: https://www.statlect.com
My background is computer science and I had a similar experience. Just a caveat: I'm not arguing that we should stop teaching theory, quite the contrary: most of the times we err on the side of the fluff. In particular, the fact that many reputable institutions are cutting formal logic, computability theory, etc. from their CS curriculums is an absolute disgrace. Intuition is hard to teach (easy to fall into the 'monads are burritos' trap) and it's something you have to work for yourself if you want to develop. My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you.
I honestly don't think it really matters what book you are studying as an introduction to a subject, as usually introductory courses are teaching well-established theory that everyone knows/agrees on. If you have no prior knowledge, a decent starting point is this: https://www.amazon.com/gp/product/1981369198/ the author's website has similar content: https://www.statlect.com
> My background is computer science and I had a similar experience. Just a caveat: I'm not arguing that we should stop teaching theory, quite the contrary: most of the times we err on the side of the fluff.
Bioinformatics at my uni is just the typical CS minus some hardware stuff + molecular biology minus some chemistry stuff; in other words, I'm pretty close to CS, too. And I share your opinion — at least for me, I don't believe things until I see them proven.
> My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you.
In addition, building an intuition can help make the understanding come faster. To give you an example, I can stare at a proof for half a day and _then_ finally get it, but one clever diagram or a descriptive commentary can save me hours of pushing through the dense text — without cutting down on the rigour (as the proof is still there). Unfortunately, it seems that maths textbooks mostly come only with the former, or the latter, but not both.
> I honestly don't think it really matters what book you are studying as an introduction to a subject
I agree that it probably doesn't matter from the content POV (i.e. the basic definitions and theorems will be there), but it could matter if we take the intuition into account.
For example, in real analysis, there's baby Rudin, but there's also all sorts of books that include all (or most of) the content, but supply it with better commentary and/or illustrations to drive the point home quicker. And I'd day that's a pretty established field, too; probably more so than stats, in fact.
Bioinformatics at my uni is just the typical CS minus some hardware stuff + molecular biology minus some chemistry stuff; in other words, I'm pretty close to CS, too. And I share your opinion — at least for me, I don't believe things until I see them proven.
> My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you.
In addition, building an intuition can help make the understanding come faster. To give you an example, I can stare at a proof for half a day and _then_ finally get it, but one clever diagram or a descriptive commentary can save me hours of pushing through the dense text — without cutting down on the rigour (as the proof is still there). Unfortunately, it seems that maths textbooks mostly come only with the former, or the latter, but not both.
> I honestly don't think it really matters what book you are studying as an introduction to a subject
I agree that it probably doesn't matter from the content POV (i.e. the basic definitions and theorems will be there), but it could matter if we take the intuition into account.
For example, in real analysis, there's baby Rudin, but there's also all sorts of books that include all (or most of) the content, but supply it with better commentary and/or illustrations to drive the point home quicker. And I'd day that's a pretty established field, too; probably more so than stats, in fact.
By the way, it seems that statlect focuses a lot on probability and so has quite an overlap with Feller. If you had to choose a primary text you'd read, which one of those two would you take?
I am so glad I am not the only one that feels this way. In high school, I didn't have to take a single probability/stats class. In college, as a CS major (!!), I had to take a single intro stats class that was completely insufficient. And when a stats education is insufficient, god damn is it insufficient. No motivating examples whatsoever (what distribution would I use to measure ${real world process}? why would I need to calculate ${X} about the distribution?), just formulas that you're expected to memorize and vomit onto an exam with no understanding of why you're doing what you're doing at all.
What is the deal with this? Why isn't stats commonly taught in school when it is by far one of the most prevalent disciplines? And why, on the rare occasion when it is taught, is it so abysmal? Statistics forms the basis for all of science, for god's sake. I've since developed a patchwork understanding of statistics on my own from various resources I've found the time to consume. For the record, I grew up in the US.
What is the deal with this? Why isn't stats commonly taught in school when it is by far one of the most prevalent disciplines? And why, on the rare occasion when it is taught, is it so abysmal? Statistics forms the basis for all of science, for god's sake. I've since developed a patchwork understanding of statistics on my own from various resources I've found the time to consume. For the record, I grew up in the US.
What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either.
I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs.
I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs.
I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
Rather than a textbook, I've had success getting a copy of the course notes directly from the stats department. The best textbooks I've read where history of statistics and philosophy of statistics.
> I’m reading Feller right now for the probability stuff, but I’m unsure about statistics.
Probability is the study of mathematical objects, and nobody is totally sure if any of them exist even in the approximate. Is anything in the universe random? The question is open, and likely to eternally remain so. Lots of things look similar to a random variable if viewed from the right perspective, but most of them aren't actually random. Not really a problem for the mathematicians, they feel no special need to study things that exist.
Statistics is roughly the study of how to deal with actual results. If you do a census, those results exist. Statisticians then need to make decisions about how to think about their results, and usually fall back on models rooted in probability. Technically speaking, "a statistic" is "any quantity computed from values in a sample". [0]
Basically, statistics is probability + data.
[0] https://en.wikipedia.org/wiki/Statistic
> I’m reading Feller right now for the probability stuff, but I’m unsure about statistics.
Probability is the study of mathematical objects, and nobody is totally sure if any of them exist even in the approximate. Is anything in the universe random? The question is open, and likely to eternally remain so. Lots of things look similar to a random variable if viewed from the right perspective, but most of them aren't actually random. Not really a problem for the mathematicians, they feel no special need to study things that exist.
Statistics is roughly the study of how to deal with actual results. If you do a census, those results exist. Statisticians then need to make decisions about how to think about their results, and usually fall back on models rooted in probability. Technically speaking, "a statistic" is "any quantity computed from values in a sample". [0]
Basically, statistics is probability + data.
[0] https://en.wikipedia.org/wiki/Statistic
Can you explain this sentence a bit more: "The best textbooks I've read where history of statistics and philosophy of statistics." ?
Are these names of actual books (Google doesn't help) or merely the themes of the stats textbooks you benefited from the most?
Thank you.
Are these names of actual books (Google doesn't help) or merely the themes of the stats textbooks you benefited from the most?
Thank you.
Not the above poster, but I concur, and recommend Jaynes' "Probability: the logic of science" for the philosophy and history, and "Breakthroughs in statistics" volumes 1 and 2 for the history as told through original foundational papers, from the 1700s on.
Probability Theory is a branch of mathematics. Statistics is the art of processing data to extract information suitable for the human cognitive system or a computer algorithm. Statistics use mathematical tools like physics or chemistry do.
Not a text, but I highly recommend the Bland and Altman Statistics Notes in the BMJ. They are usually 1 page, easy to read explanation on a single statistic topic.
Here is one on the Odds Ratio for example https://www.bmj.com/content/bmj/320/7247/1468.1.full.pdf
Here is one on the Odds Ratio for example https://www.bmj.com/content/bmj/320/7247/1468.1.full.pdf
> I don’t even know what the relation between probability and statistics is
That's a great question, and I think the lines are more than a little blurry.
My attempt at an answer would be:
Probability: Given a set of dice and coins and an order for rolling and throwing them, what is the chance of a specific outcome?
Statistics: Given a set of outcomes, what dice where rolled?
So if you want to know if smoking kills, you tally up medical history, and use statistics to see if there is a relationship between smoking and dying.
If you want to know the probability of smoking killing you, you look at the risc each cigarette brings to the table and tally it up using probability theory.
More elegantly phrased examples can be found on Stack Overflow: https://stats.stackexchange.com/questions/665/whats-the-diff...
That's a great question, and I think the lines are more than a little blurry.
My attempt at an answer would be:
Probability: Given a set of dice and coins and an order for rolling and throwing them, what is the chance of a specific outcome?
Statistics: Given a set of outcomes, what dice where rolled?
So if you want to know if smoking kills, you tally up medical history, and use statistics to see if there is a relationship between smoking and dying.
If you want to know the probability of smoking killing you, you look at the risc each cigarette brings to the table and tally it up using probability theory.
More elegantly phrased examples can be found on Stack Overflow: https://stats.stackexchange.com/questions/665/whats-the-diff...
I kind of like M.G. Bulmer's "Principles of Statistics". It's short and to the point so there's a chance of getting through it all. I really like the discussion of distributions in terms of raw data, it makes thinking about mean, variance, higher moments etc., much easier. It also doesn't skimp on the mathematical theory, but it doesn't allow itself to get bogged down by it.
That said, there's a chance I just read it late enough in my career to be more ready for its content.
That said, there's a chance I just read it late enough in my career to be more ready for its content.
I cannot recommended them because I have them on my back burner, but I would like to start working through the "Think" Stats series soon [1].
I love the premise: "if you know how to program, you can use that skill to learn other topics."
Perhaps someone here can speak to their experience with some of these books?
1: http://www.allendowney.com/wp/books/
I love the premise: "if you know how to program, you can use that skill to learn other topics."
Perhaps someone here can speak to their experience with some of these books?
1: http://www.allendowney.com/wp/books/
I am doing harvard 110 now (free lectures on youtube) and I am working through his book Introduction in probability.
Maybe it is too basic for you, but It is focused on the intuition part and I can recommend it!
Maybe it is too basic for you, but It is focused on the intuition part and I can recommend it!
Probability & statistics by de Groot is the standard text I believe. Full of examples and questions.
So so cool ! And it goes to show how poorly probabilities and statistics are usually taught, it's such a waste. I'm working on a non profit project aiming in parts to aggregate this kind of pedagogical tools into a collaborative learning map and serving it in a personalised way: https://sci-map.org. Early phases still, but if people are interested to contribute please hit me up!
sci-map sounds very interesting. Have you looked at metacademy.org before?
They did a lot of good work on the data model (concepts, resources, learning pathways, etc), and also collected a lot of content, mostly on computer topics.
https://metacademy.org/graphs/concepts/bayesian_logistic_reg...
Sadly the project is no longer actively developed but if you haven't seen it yet, you should definitely check out for inspiration: https://github.com/metacademy
Sadly the project is no longer actively developed but if you haven't seen it yet, you should definitely check out for inspiration: https://github.com/metacademy
I didn't know about, seems very similar indeed! I will check it out in details, thanks a lot for sharing !
"If you roll 2 six-sided dice, what are the chances you roll at least one dice above 5 (5 or 6)?"
A nice trick to visually solve this in your head I heard once is:
If you think of rolling two dice as a square. X and Y are each dice. You get a 36 square board. Getting 1 six is just the upper boarder. 6 on the top, 6 on the right (6 and 6 overlap). So 11 out of the 36 squares.
A nice trick to visually solve this in your head I heard once is:
If you think of rolling two dice as a square. X and Y are each dice. You get a 36 square board. Getting 1 six is just the upper boarder. 6 on the top, 6 on the right (6 and 6 overlap). So 11 out of the 36 squares.
Chances of rolling below 5 are 4x4/36 = 16/36, so above are 20/36
And here is the board: https://www.edcollins.com/backgammon/diceprob.htm
And here is the board: https://www.edcollins.com/backgammon/diceprob.htm
There is the small matter of "equal to 5"
Another way to think about it using the square board concept would be to figure out how many ways you can get not the result you're looking for, and take 36 minus that number for the number of possible squares out of 36 squares. So getting "no 6's" on either die would be the square on the board of 1 through 5, by 1 through 5, or 25 squares. So inverting that we'd arrive at the 11 squares.
In studying probability, I found that accounting for the "overlap" as you described it was more tedious in more complicated problems than just always calculating the joint probabilities and inverting them.
In studying probability, I found that accounting for the "overlap" as you described it was more tedious in more complicated problems than just always calculating the joint probabilities and inverting them.
This looks brilliant. This can be very useful to grasp the concepts of probability and statistics in a visual way. I've been struggling to understand some of the concepts and I hope to use this as a supplement. Although, I don't believe it can replace a university course or a proper text book.
[deleted](1)
How come that an undergraduate person (at the time) makes one of the most compelling statistic textbooks?
Is it because there are many more amateur statistic textbooks in existence, or published attempts at one (so more chance for a runaway success to be picked up)?
Or is it because people in the statistic textbook industry don't feel this frustration and/or don't dare to take any risk?
Is it because there are many more amateur statistic textbooks in existence, or published attempts at one (so more chance for a runaway success to be picked up)?
Or is it because people in the statistic textbook industry don't feel this frustration and/or don't dare to take any risk?
nice use of interactive data visualization to explain concepts, would love to see more teachers use this approach.
I must say that this website/way of reading a book is simply gorgeous
Brain does not count. It updates/alters its structures - the process which humans model with numbers and back propagation.
Numbers do not exist outside of human cultures.
Numbers do not exist outside of human cultures.
Got stuck at chapter 2 since there was zero indication what a valid syntax for a set was.
What a beautiful study resource.
This is a really cool website!
Very interesting and helpful !
Amazing. Thanks for sharing!
Gorgeous web design
With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for trajectories, acceleration and distances (derivatives and integrals), areas, volumes, amounts, etc. But on probability your gut-feel will always fool you.
In the end, you see a handful of math bloggers bemoaning the lack of education in probability and the nonsense being discussed by journalists and politicians. And it hardly matters whether it's an election or a pandemic. The lack of understanding of uncertainty and the false belief that one can reason about these without looking at the numbers too closely is dangerous.
Sorry about the rant. But...
Dear creator of seeing-theory.brown.edu, if there is one thing you could change about the project to make it different and infinitely more useful: Please start the first chapter with the goat problem[1], then go through a couple of examples from chapter 10 in Thinking Fast and Slow[2], the discuss information (maybe with a simplified version of Mendel's pea experiment[3]), discuss distributions and leave expectations and variances for much-much later.
[1]: https://en.wikipedia.org/wiki/Monty_Hall_problem [2]: https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow [3]: https://www.sciencelearn.org.nz/resources/1999-mendel-s-expe...