Comment by lordnacho | Hacker News Reader

Comment by lordnacho | Hacker News Reader

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I second that. Stats is so much more applicable, and it is also the area where common sense will fail you the most. Simple things like birthday paradoxes, Gambler's fallacy, Monty Hall problem, and so on are all contrary to common sense yet not taught in schools.

And you need stats all the time. Every time someone says crime has never been worse, you need to know how to decide if you believe it. When someone says flying is safer than driving, you need to know how to compare. A lot of stats thinking is also not formulas, but qualitative things like "Am I comparing apples to oranges".

Calculus by contrast is pretty common sensical. Acceleration is the derivative of velocity, you find the local extremes by finding the zeros of the derivative, the integral of a net shaped field is the rim... that all makes sense with some pictures.

I also haven't come across it much in finance, apart from derivatives work, where funnily enough you can't do with high school calculus, but you need stochastic calculus. Which contains some quite statistics related concepts.

I suspect calculus is less commonsensical to most people than you think :-) That said, I've argued the parent's point for years. Basic stats/probability/economic biases are really fundamental stuff and don't require a lot of complicated math if you don't want them to.

There are some basic calculus concepts that are useful. Area under a curve. Basic derivatives (for finding maxima/minima, etc.). But the truth is that I've rarely used calculus outside an academic setting and I use at least basic statistics/probability thinking all the time.

I think the reason calc is preferred, to prob/stats is because the intuition is much more visual than the latter. Furthermore, students up until that point spend the majority of their time studying various functional forms (conic fxns, exps/logs etc.) Functional analysis seems like an obvious next step.

Though stats/prob are perhaps more 'useful' in non-academic life. Imparting the notion of what probability actually is and how Bayes' rule fits into it requires a very gifted teacher with a fundamental grasp of the real mathematics and on top of that the ability to communicate it and link it to previously learned material aka algebra/ pre-calc. You can make it visual, but it would seem to require defining the concepts of sets, sampling, distributions etc.

The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it. As such, most students turn into machines calculating things as opposed to viewing math as a way to reason.

>The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it.

Yes. I agree generally--often even at relatively advanced levels.

The stats course I took in engineering grad school was so heavy on the math that I'm not sure I appreciated what the math was in service of. By contrast when I took a less math-heavy stats course later I much better appreciated the underlying principles.

But yeah, it's very easy to turn any of these math-related topics into: memorize the formula, plug the numbers, come up with a result.

In the UK high school math is really a preparation for a hard science/engineering degree, and there's a lot more calc than stats in hard science/engineering.

That's why calc is central. I wouldn't be surprised if it's the same in the US.

This means most people have no idea what the math they learn is used for, because they never reach a level where they have to use it.

So they leave school with the idea that math is annoying bullshit that they don't understand and can't use.

I think it would be better to stream math by ability much earlier. Make everyday math compulsory for everyone - interest rates and loans, basic stats, currency rates, areas/volumes, simple trig, maybe some slightly more advanced finance, perhaps touch on some calc ideas.

Only teach calc and scientific math to those with the talent and interest for it. Give everyone else a taste so they understand a little, but don't try and get them ready for a career they'll never go into.

Having taken hard science in university/ during my PhD, it's hard for me to say that calc presents itself more than stats. Perhaps in engineering but only in the early years of uni, physics same thing- year two is an exposition to quantum mechanics... In chemistry and biology I would say stats/prob definitely predominates. In chemistry calc begins to appears in physical chem.

Why most people don't understand the math they learned and what it is used for, I would argue has much more to do with how they are not taught to interpret what they see around them mathematically- granted that takes years of practice, but we do not care to teach this to children.

Now streamlining math is all well and good, but we may loose some of the greatest mathematical minds by doing so. Many famous mathematicians (Euler, Lagrange for example) were slated to be someone else- professionally that is. And it took other great mathematicians to notice them.

As if algebra or calc was common sense. I did learn about the Monty Hall Paradox but I'm still doubting it

The Monty Hall problem is about being given two sets of information. First they give you a 33% shot, then they give you a 50% shot. Take the 50% shot.

No, the second chance is supposed to be better than a 0.5 and I could outline the argument, but I don't trust it.

The key is that Monty Hall is guaranteed to show you a losing door. If that weren't the case -- if he just picked one of the remaining doors at random -- then the fact that he happened to pick a losing door would count as evidence that your door is a winner, leaving you with a 50/50 shot. I think that's one source of intuitive discomfort, and even of lasting confusion if the scenario isn't specified precisely.

That's not true. It's a 66% shot. It has to be, because there only two options, and (we agree) one of them was 33%, so they other must 66%.

Consider 3 doors: A, B, C. If you pick the right one you win the prize. Say you pick door A. Monty Hall opens B or C and reveals that there's nothing behind. Now he asks if you want to stay with A or switch to the closed door. What should you do to maximize your chances of winning?

Right before any of the doors is opened we have no idea what to expect, so there are 3 equaly likely possibilities:

1. prize is behind A 2. prize is behind B 3. prize is behind C.

In case (1) you win by staying with your choice of A. In case (2) Monty Hall must open C (by the rules), so you'll win if you switch to B and in (3) the host will open B(by the rules) and you should switch to C.

In 2 cases out of 3 (2/3) you win by switching and in only one case out of 3 (1/3) you win by staying with your first choice. Since 2/3 > 1/3, you should switch.

This problem is easy if you know the defintions of sample space, event and the Equally Likely Probability Formula.

I prefer thinking about the 100 doors case.

You are shown 100 doors 1 ... 100. Imagine you pick 1 door.

Of the other 99 doors Monty tells you which 98 of them don't have the prize.

He now asks if you want to switch to the 99th door that he didn't cancel out.

Intuitively that seems to make much more sense to me.

Yes, indeed it did for me, as well, but then I'm not trusting my intuition that failed me for the three door version, and I'm desperately trying to prove it wrong, because I'm biased by my first commitment to an answer.

I think that commitment bias or what it's called is also an explanation for the misunderstanding in the first place. The argument against the increased chance, as I would use, works both ways and that is much more apparent the smaller the initial chance is. But I only intuitively used it to defend my position.

Just in case....

A sample space is the set of all possible outcomes of a random experiment. An event is a subset of a sample space.

Let S be the finite sample space with equally likely outcomes in it. Let E be an event. Then the probability of E is the number of outcomes in E over the total number of outcomes in S.

Now consider 52 cards in a deck. The sample space of outcomes here are the 52 cards. What is the event that the chosen card is a black face card? E = {J-club,Q-club,K-club,J-spade,Q-spade,K-spade}. What is the probability that the chosen card is a black face card? 6/52 by the formula above.

Monty Hall problem can be solved by using the above as a guide.

Sample space here : {switch, switch, stay-with-your-original-choice}. By the formula above, the probability of switching is 2/3 and the probability of staying put is 1/3.

If you're doubting the formula, I can provide you with a proof.

As I mentioned in another comment, it's important to note that Monty Hall is guaranteed to open a door with a goat behind it (a losing door). If he chose randomly and just happened to reveal a goat, the answer would be different.

Indeed. This is the major reason I don't like the Monty Hall problem. When using stats and probability (at least in science), the universe is not your adversary.

I think there is an easy and intuitive explanation:

Two thirds of the time, you pick the wrong door on your first choice. So far, so obvious. Now, you picked a wrong door, Monty opens the other wrong door, and there are only three doors. Obviously the remaining door has the prize.

Every time you pick the wrong door, and then switch, you win. That means two thirds of time, if you switch, you win. That makes switching the right strategy.

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