Which of these sockets are in the right place? (If any)

[morqthana]

But hey, maths is not a perfect science,

https://www.reddit.com/r/NoStupidQuestions/comments/18896hw/my_sons_third_grade_teacher_taught_my_son_that_1

 

[morqthana]

I've read dozens, if not hundreds, of explanations' in my time, as have many statisticians, but that doesn't alter the fact that most of us still have grave 'intuition problems'! There's also at least one PhD thesis about it knocking around!

But this seems intuitive:

You pick a door.

There's a ⅓ probability that the prize is behind your door, and a ⅔ one that it is behind one of the other two doors.

Those probabilities cannot be changed by the show's host opening one of the doors you didn't pick. They just cant.

So the unopened door now has to take all of the ⅔ probability that the prize is behind one of the other two doors.

As I wrote, the worry is that since what many experts regard as the 'obvious answer' is actually incorrect, they naturally worry about the correctness of other assertions they make about things which 'seem obvious' to them.

That sounds like a good thing.

 

[JohnW2]

But this seems intuitive:
You pick a door. .... There's a ⅓ probability that the prize is behind your door, and a ⅔ one that it is behind one of the other two doors. .... Those probabilities cannot be changed by the show's host opening one of the doors you didn't pick. They just cant. .... So the unopened door now has to take all of the ⅔ probability that the prize is behind one of the other two doors.

I'm not going to go into all this again now but, yes, that sounds pretty intuitive, but it gets more complicated when one starts to think (perhaps 'too much') more deeply, not the least because it becomes quite easy to end up feeling (incorrectly) that the correct answer flies in the face of some basic truths of probability theory.

That sounds like a good thing.

Up to a point, it is. However, it continues to be a concern, particularly to those who have not yet come to fully understand why their 'intuitive' answer Monty Hall is not correct.

 

[morqthana]

Such people are precisely the ones who should be cautious about going what seems "obvious" or "intuitive" to them.

 

[JohnW2]

Such people are precisely the ones who should be cautious about going what seems "obvious" or "intuitive" to them.

They probably are, but there's quite a lot of them.

Many moons ago, one of my colleagues seriously considered giving up doing consultancy work because of this (and. instead, 'retire early'!). He argued, quite reasonably, that to give professional advice conscientiously and 'decently', required that he had confidence in the fact that things that he sincerely believed to be correct truths were, indeed, correct truths - but that his experiences with Monty Hall had brought him to doubt whether that would necessarily always be the case. However, when he thought a bit more, and particular when he came to realise that he had quite a lot of 'good company', he became more 'pragmatic' about things.

 

[ebee]

If I remember correctly, didn`t a prof of maths write into the show that the producers statements about doubling the chances by changing their minds was rubbish, only to write in again the following week with the opposite view?

 

[JohnW2]

If I remember correctly, didn`t a prof of maths write into the show that the producers statements about doubling the chances by changing their minds was rubbish, only to write in again the following week with the opposite view?

That wouldn't surprise me at all, since that was the initial ('intuitive') reaction of very many experts in the field, including many professors of maths and statistics etc.

Morqthana gave a very concise and simple explanation of the reasoning behind the correct answer. He called that reasoning 'intuitive', but if that really was his initial thought about the issue, then that would make him pretty exceptional, 'superior' to many of those professors etc.

The problem arises because those very professors will be drumming into their students the fact that if the prize is put, at random, behind one of three doors, with equal probability that it is behind each of the doors, then the probability of it being behind any particular door is 1/3 "...and nothing can change that". If one takes that view (and it becomes part of one's 'intuition'), then one will conclude that there will always be the same probability (1/3) of the prize being behind the contestant's chosen door and the one they are invited to 'swap to', regardless of anything else that has happened (i.e. the host having opened a door).

The issue is that that "...and nothing can change that" (which is essentially what is always taught, usually 'drummed in') is not strictly true. The probabilities can change IF 'new information' is brought to bear on the situation - in this case by the host being constrained to open a door with the knowledge that the prize is not behind it - thereby, as morqthana said, shifting all of the 2/3 probability into the door that the host didn't open.

However, that then leads many to another 'thought problem'. If they arrive on the scene after the host has opened a door, all they see is two doors, one of which has the prize behind it, and therefore tend to 'subconciously assume' that the probabilities of the prize between those two doors being equal (i.e. 1/2 each)

 

[ebee]

Hit nail on head there as usual John,
Yes I could see the change from 1 in 3 (1/3) to 1 in 2 (50% of the original 33.33%) thereby being an increase of 50% of the chance original choice by now having to select 1 from 2 rather than 1 from 3 and I suspect the Prof thought similar at first. it did vex me so after a few days i gave up on it. a week or two later with no thought about it i had a Eureka moment and it suddenly fell into place.
a strange thing a human mind is - well mine is, anyway.

But I was not thinking about it, I was having a shave at the time - unless looking at the mirror and i thought "Oh Ebbe, you look like an old goat! and it made me think)

 

[JohnW2]

The only mention in this years GSCE tuition regarding decimal fractions, as I've seen it, has been solely regarding fractions containing powers of 10, IE: 1/10, 10/40 etc .....

As perhaps concluding comment in relation to this discussion about terminology, I don't think anyone (including myself) has yet pointed out how irrelevant the discussion is to almost anything 'practical'.

I'm sure that all of us here, and the majority of people in general, understand, and can 'use' the various ways in which non-integer ('not whole number') numerical values can be represented [ e.g. 0.8, 4/5, 8/10, 80/100 etc. or 6.8, 3/4, 68/10, 680/1000 etc. ] but, in the vast majority of situations it does not matter a jot what one calls those representations. About the only 'relevance' I can think of is the issue raised by Sunray, namely in relation to the wording of exam questions etc.

I would also observe, as I have been discussing off-list, is that some of the apparent confusions (and speculations of changes over time) may well be due to UK-US differences, with some US terminology having perhaps crept across the water over time?

 

[opps]

Potentially mind-boggling, but extremely useful, even in electronics

I initially encountered them on my economics degree. From memory, they were pretty important in the development of AC power distribution.

 

[JohnW2]

I initially encountered them on my economics degree.

I must say that I'm struggling a bit to think how imaginary/complex numbers would come into economics

From memory, they were pretty important in the development of AC power distribution.

Yes, other than for 'academic maths' interest, I think many of the main practical applications are in relation to electrical engineering, particularly anything to do with phase differences between two AC waveforms, since they enable both magnitude and phase of a waveform to be represented by a single expression. They also have a lot of use in relation to quantum mechanics.

 

[SUNRAY]

Flying visit by my daughter to pick something up and she dropped in these 2 revision sheets for GCSE

and the answers

Thes are the sheets downloaded from a schools server and printed as work sheets for GCSE revision purposes, until assisting my grandson I had learnt at college decimal fractions were common fractions with denominators of powers of 10 and before that at school in 1970s there are no such thing as decimal fractions.
Since this thread started I've also spoken to another school student at the end of lower 6th who had learnt the same (Decimal fractions are common/vulgar fractions with powers of 10 above or below the line) but in 6th form retaught as just the denominator and given this site as a point of reference: https://www.splashlearn.com/math-vocabulary/decimals/decimal-fraction

Until this thread I had never known the term decimal fraction to refer tor a non integer, only ever refering to powers of 10 fractions.

Am I surprised there is confussion when there seems to be so many definitions?

 

[JohnW2]

Until this thread I had never known the term decimal fraction to refer tor a non integer, only ever refering to powers of 10 fractions.

As I've said, even the Oxford Dictionary, today, agrees that "fraction" simply means non-integer, so the main debate/confusion is about how qualifying that with "decimal" changes things. You (well, GCSEs) seem to now regard it as meaning that it only refers to a representation as 'numerator over denominator" AND only if the denominator is a power of 10, whereas the traditional meaning I (and I presume also morqthana, since it was what he wrote that started all this!) was taught (a long time ago) was that it merely meant that the non-integer was expressed using decimal notation (e.g. 123.456).

However, as I've said, it doesn't really matter a jot what words we use, other than to those who write exam papers etc. and those who are expected to be able to understand what they have written. No-one will (hopefully ) "get any sums wrong' as a result of these 'word issues'

 

[opps]

I must say that I'm struggling a bit to think how imaginary/complex numbers would come into economics

TBH I cannot completely remember. I think it was something to do rates of depreciation. Rates of depreciation was why I had to learn advanced calculus/partial derivatives. Something I struggled with at A'level, but excelled at uni where all I had to focus on was integration and matrices on that particular module. I walked out of the exam and "calculated" that I had 96 or 97%. I got a first on that module. Shortly after I had to meet the head of school for a lack of attendance. Most of my module scores were pretty unimpressive but when he looked at my results for that module, he said "well you clearly aren't stupid". I was a lazy effer, he stopped me being kicked out of uni. I never got to thank him. He is Rodger Leigh (Middlesex University)

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www.researchgate.net

And if any of his family members stumble upon this post whilst looking for him. He was a lovely fellow. I am glad that switched from a BSc in economics to a joint major in Economics and Geography. The poly/uni taught us how to learn. The economics school didn't care about grammar or syntax. The geography school (largely economic policy in practice) insisted that essays be readable, cited and coherent.

Meh. I decorate for a living. But I was lucky enough to be able to be go to uni and learn how to learn. Schools, and many universities teach via rota.

Oh and whilst I am at it, I want to thank Aram Eisenschitz.

I missed the first 3 of his lectures. He sent me a letter to "remind" me that the subject was not a distance learning subject and that unless I started attending he would contact my local educational authority. I thoroughly enjoyed his subject (a marxist critique of capitalism)

 

[morqthana]

Morqthana gave a very concise and simple explanation of the reasoning behind the correct answer. He called that reasoning 'intuitive', but if that really was his initial thought about the issue, then that would make him pretty exceptional, 'superior' to many of those professors etc.

Well....

But no - what I meant was that you don't have to be an expert in probability, or do any complicated maths, to see how that explanation makes sense.

FWIW, my initial thought, when I first encountered the problem, was that swapping to the other door made no difference.

The problem arises because those very professors will be drumming into their students the fact that if the prize is put, at random, behind one of three doors, with equal probability that it is behind each of the doors, then the probability of it being behind any particular door is 1/3 "...and nothing can change that". If one takes that view

I don't see how anyone can not take that view.

(and it becomes part of one's 'intuition'), then one will conclude that there will always be the same probability (1/3) of the prize being behind the contestant's chosen door and the one they are invited to 'swap to', regardless of anything else that has happened (i.e. the host having opened a door).

The starting position is a set of three doors, and the probability that the prize is behind a door in that set is 1. After choosing a door there are then two sets of doors. The chosen one, and the unchosen ones. At all times the probabilities have to add up to 1.

The chosen door has a probability of ⅓ and the set of unchosen ones has a probability of ⅔. That ⅔ is ⅓+⅓.

If no door is opened by the host then swapping would not change the probability of the contestant having picked the right one originally or ending up with the right one.

If one of the unchosen doors is opened that too cannot change the probability of the contestant having picked the right one originally, but it does change how the ⅔ probability in the unchosen set is distributed amongst the doors.

The issue is that that "...and nothing can change that" (which is essentially what is always taught, usually 'drummed in') is not strictly true.

Yes it is, in the context of this scenario. Remember the "that" are the probabilities after the contestant has chosen, which are that there's a 1 in 3 chance the prize is behind his door and a 2 in 3 chance it's behind one of the other two doors.

The probabilities can change IF 'new information' is brought to bear on the situation - in this case by the host being constrained to open a door with the knowledge that the prize is not behind it - thereby, as morqthana said, shifting all of the 2/3 probability into the door that the host didn't open.

Opening a door does not sometimes cause the prize to move, and therefore does not alter the probability of the prize having been behind the contestants door at the start.

However, that then leads many to another 'thought problem'. If they arrive on the scene after the host has opened a door, all they see is two doors, one of which has the prize behind it, and therefore tend to 'subconciously assume' that the probabilities of the prize between those two doors being equal (i.e. 1/2 each)

That's a different scenario - the contestant is choosing between 2 doors, not 3, so the probabilities become ½ & ½.