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Comment by mcprwklzpq | Hacker News Reader

Programming does definetly helps me when i do not understand an idea. For exmaple:

I did not understood why in the Monty Hall problem [1] do i have 2/3 chance of winning if i do change the door instead of 1/2 that my intuition tells me until i wrote it as two functions.

function change() {return random(3) != random(3);} // 2/3 chance

function not_change() {return random(3) == random(3);} // 1/3 chance

Then i also saw that i do get that 1/2 chance of winning if i choose between changing the door and not at random.

function random_change() {if (random(2) == 1) {return change();} else {return not_change();}} // (1/3)/2 + (2/3)/2 = 1/2 chance

This was so much more convincing than math and theory and even explanations with cards and drawings because i could run it in a loop a 1000 times and see the results be close to the prediction.

1 - https://en.wikipedia.org/wiki/Monty_Hall_problem

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Really? I also value the evidence that numerical simulations provide, and if I didn't know of the problem at hand, the simulation would indeed convince me of the correct answer. But has one really understood something just because one has become comfortable with its behavior?

I would say it was more so the process of implementing that same logic forces you to understand it, and then seeing that logic pan out is a confirmation of what was learned in implementation.

Much like you can understand the math of a lever and see levers working, but precalculating that x can lift y with a z long lever, and then building it, reinforces the logic you used in precalculation and binds the abstract to the real.

I was able to predict how simulation would go when i saw the program that i wrote. When i saw the simulation confirm my prediction it was so much more convincing then any other method. If the logic was to complex for me to understand then i would only see the behavior and not much more.

The Monty Hall problem is typically not clearly explained with the confounding precondition that the game show host won’t open a door that reveals the prize. Once I understood that there was a bias against one of the doors, the problem became obvious.

I agree. I think a lot of these probability puzzles end up being "illusions of emphasis", ie., that we fail to grasp the problem because how components of it are expressed in language.

When the host opens the door and why need to be repeated several times and with lots of emphasis before asking the question. These ordinarily minor details are extraordinarily relevant, and the language used should express that.

Yes, because if the problem was expressed purely formally from the start, there could not possibly be any controversy. The appeal of Monty Hall is precisely this interplay of natural and formal language.

People aren't generally very good at probability. Plenty of people get it wrong despite understanding the puzzle correctly.

This is a fairly common reacton to the puzzle these days.

In the time and place of the problem's first widely-distributed posing (Parade Magazine, 1990) it might reasonably be assumed that many people seeing it there were somewhat familiar with the game show it is loosely based on.

Even without that background, one might reasonably deduce that the second door opening never reveals the prize, as there would be no point, in that case, in asking whether the contestant wants to change her pick.

That is a somewhat meta argument, in that it involves understanding human intent and what makes for a good puzzle, and it means that it is not just a math/logic/probability challenge, but it is nevertheless a good puzzle. There's no law that says a puzzle must explicitly state every fact of the matter.

I assumed that the game -- at least the math problem -- starts when Monty has opened a door and revealed a goat.

The point is if Monty knows he’s going to reveal a goat or if he just does so by chance.

It doesn't matter if there's a bias against one of the doors if you're only considering the cases where the door Monty opened did happen to reveal a goat. In two thirds of those cases, you can improve your choice by switching. It doesn't matter if Monty knew the goat was there ahead of time or not!

I was going to make a lengthy logical argument, but I started to wonder if maybe I was reasoning incorrectly as I tried to prune the redundant cases, so instead I took two minutes to write this Python program to enumerate all 27 equally probable possibilities; if you pipe its output to sort | uniq -c you will see that I am correct:

    xs = [(you, monty, car) for you in range(3)
                            for monty in range(3)
                            for car in range(3)]
    print(xs)
    for you, monty, car in xs:
        if monty == car:
            print("Monty revealed the car, too bad")
        elif you == car:
            print("You win only if you do not switch")
        else:
            print("You win if you switch")

Now I realize this is dumb because I wasn't excluding the cases where Monty opens the door you picked. Excluding those cases takes us back to 50:50! So I guess it really does matter that Monty knows in advance where the goat is...

How embarrassing that it took me two minutes to get the wrong answer and two hours to realize it — just long enough that I couldn't delete my confident assertions above :) This probably forms some kind of evidence about the quality of evidence you can get out of computer experiments without careful thinking :)

I think if I was not aware of the Monty Hall problem and I ran that simulation, I'd assume there was a bug in my code.

The trick to understanding the Monty Hall problem isn’t to run it 1000 times, it’s to imagine it with 1000 doors. Thinking about the problem with an arbitrarily large number of doors make the solution intuitively obvious. No code required.

Yes! This was how I first figured out how to understand it, and how I explain it to anyone if it comes up. It's immediately obvious with a large number of doors.

I pick 333 doors, and Monty opens 333 doors?

No the variation with 1000 doors is that you open 1 door, Monty opens 998 doors and asks if you want to switch.

Yeah, the use of 3 doors, and the bit where one is opened just kinda hides what’s happening. Which is that you’re actually choosing between one random door, or ALL the other doors.

The Eureka moment for me was similar, but slightly different. Consider a variant game with N doors. You pick a door, then Monty offers you two switch your payout to "the best single result from the remaining N-1 doors". Clearly, you would swap.

Now set N to 3, and consider that, by opening a door without the prize, that's exactly what Monty's offering you.

This trick did not help me. Intuition told me in the end i have to choose between 2 doors and the chance of winning is 1/2. And it is if i choose in the end at random. This is what made the question confusing for me.

The opening of doors prior to having you make your decision is a red herring. The person opening the doors knows which one the prize is behind, and they’re only going to open doors which have no prize. So if there’s 1000 doors, you have a 1/1000 chance of choosing the correct door, and there’s a 999/1000 chance that the prize is behind a different door. So if the host selectively opens 998 losing doors, leaving your original selection and one other door closed, there is still a 999/1000 chance that the prize is behind the door you didn’t choose. If the host instead opened the doors randomly, there would be a 998/1000 chance that they’d reveal the prize before narrowing it down to the two doors. But they don’t open them randomly, which is why the door opening doesn’t matter. You’re effectively being given a choice between the one random door you originally chose, or every other door.

This was not what bothered me. I needed to see how this puzzle does not contradict my intuition that if i flip a coin to choose between two last doors i do get 1/2 chance of winning whatever are the chances for each door. So with 1000 doors: (1/1000)/2 + (999/1000)/2 = 1/2 chance of winning. And in general (A+B)/2 = 1/2 if A+B=1.

If you flip a coin at the decision stage, then you would have a 50/50 chance of winning. But that approach would mean you’re disregarding information that you can use to increase your chances of success above 50/50. When you make your initial choice you have a 1/1000 chance of guessing correctly, meaning there is a 999/1000 chance that the prize is behind another door. By eliminating 998 doors from contention, you’re not changing those odds. There is still a 999/1000 chance for the prize to be behind a door other than the one you initially picked. Only now, instead of having 999 doors to choose from, you only have one. So by switching your choice, you increase your chance of success from being whatever it was initially, to being (1 - whatever it was initially). Which will always be greater than 0.5.

This is materially the same as being given a choice between opening one door randomly, or being able to open every other door. Because the conditions of opening doors prior to the final choice is that the door you initially picked can’t be opened at this phase, and neither can the winning door.

I had the same experience with that very problem. Didn't understand it before I programed it.

Maths is not experimental. This roundabout, handwavy, vague way to deal with things causes more harm than good, in my opinion. Try to understand a proper mathematical proof instead.

I disagree. Math is very experimental - it is bottom up science where you first touch various details in the dark and generalization comes later.

I am with Arnold on that.

> Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well.

https://en.m.wikipedia.org/wiki/Vladimir_Arnold

Math is experimental at its roots. If the result here wasn't usable in real life, scarcely anyone would develop that specific branch of probability theory.

Instead they would think of adjusting the axioms and work out a useful probability theory.

For example, most children check 1+2 experimentally before accepting the theory.

What’s the theory to accept for 1+2?

I think he means loosely that the theory here, is that they add up to three. Kids will grab objects and group and ungroup them while counting and so on, before outright accepting math is real.

That can be trivially inferred from 110-643 (Vol. 2, p. 86) in Whitehead & Russell's Principia Mathematica.

Not the GP, but this is my take on their comment. Children learn counting before they learn an algorithm for addition. The "theory" to accept here is that counting a set of size x, then continuing to count, starting from x, a set of size y, yields the same result as the addition algorithm for x+y that they learn.

I consider myself borderline math illiterate. I understand the monty hall problem perfectly well through means other than hard math proofs.

So, clearly, understanding the proof isn't required. It is a great deal more work than many other methods of understanding the problem, so why prescribe it as the best way?

You might be comfortable with it, but not everyone has the time nor inclination to become mathy enough to live life through your lens.

Here's a glimpse that may change your mind about mathematics as an experimental science:

Mathematics: An Experimental Science - https://www.math.upenn.edu/~wilf/website/Mathematics_AnExper... (PDF)

> A computer is used by a pure mathematician in much the same way that a telescope is used by a theoretical astronomer. It shows us "what’s out there."

> From such explorations there can grow understanding, and conjectures, and roads to proofs, and phenomena that would not have been imaginable in the pre-computer era. This role of computation within pure mathematics seems destined only to expand over the near future and to be imbued into our students along with Euclid’s axioms and other staples of mathematical education.

Also:

Turtle Geometry: The Computer as a Medium for Exploring Mathematics - https://mitpress.mit.edu/books/turtle-geometry

>Mathematics: An Experimental Science

That was an amazing read. It certainly made me rethink my position. Thanks!

This roundabout, hand-wavy, vague way to deal with things is how mathematicians solve problems. You have to have an intuitive understanding of what's going on before you can formalize the solution into a proof.

I majored in math in undergrad, and one of the required courses was about communicating mathematical ideas. One of my bigger projects in that course was to explain the Monty Hall problem so it could be understood by the general public, and I got a good grade by putting together a lesson plan that included this type of simulation.

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