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

[SUNRAY]

Yeah - just going over why the speed appears to have changed.

lack of smog

 

[JohnW2]

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. ...

Indeed, once one has been exposed to that 'explanation' (or has thought it up for oneself).

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

I suppose it's fairly reassuring to know that, like the rest of us, you are also human

I have to say, that in all the countless discussions about this that I've been involved with or aware of over the years/decades, I'm not sure that I've ever come across anyone, even undisputed experts in probability theory, who have claimed that they immediately knew the correct answer when they first came across the 'question'!

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

Quite - but it depends upon what you mean by "that view". The fact that probabilities cannot change due to subsequent events is drummed into students very strongly from Day 1, but seemingly without the necessary qualification that it ceases to be the the case if someone/something 'moves the goalposts' (which can 'move probabilities around') - and that will usually (always?) require at least partial knowledge about 'the answer'.

In the case we are discussing, that moving of the goalposts results from the fact that the host is constrained to only open a door which doesn't have the prize, thereby, as you've said, transferring all of the 2/3 probability to the 'third' door. IF the host merely choose a door ('at random', without knowledge of what was behind in) to 'take out of the equation' (but didn't open it), hence leaving the contestant with just a choice between their initial choice and the 'third' one, then the probability of the prize being behind the third door would, indeed, still be 1/3, so it would make no difference whether or not the contestant changed their choice. The point here, of course, is that it would remain possible (with 1/3 probability) that neither of the doors available to the contestant had the prize (i.e. if the prize was behind the door which the host had 'taken out of the equation').

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.

Indeed, as above.

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.

True, but the problem here is that probably most people, particularly 'the general public', do not think about probabilities 'that way around' - e.g. they think about the probability that their chosen number will win the Lottery, not the probability that one of the numbers they have not chosen will win.

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.

Again true - but, of course, the 'problem' arises because many people will (incorrectly) think/say exactly the same about each of the other two doors (i.e. forgetting that the host has revealed the (zero) probability of the prize being behind one of the doors).

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

Quite so - but, as I said, many people will (incorrectly, and probably subconsciously) assume that, if the prize was allocated to a door 'at random', that in the 2-door scenario, there will be an equal (1/2) probability of it being behind either of the doors.

As I hope you understand, I am in no way disagreeing with the correct answer, nor with the ('simple') explanation you have provided that it is correct - but, rather, am indicating some of the reasons why so many people (including many 'experts') Have so much of a problem with this!

 

[JohnW2]

Yeah - just going over why the speed appears to have changed.

The numbers which ebee and yourself are talking about are, of course, the speed of light in a vacuum. The speed of light in air is very similar but it is much slower in media such as water and glass (I think generally about 70% of the speed in a vacuum).

However, it's perhaps worth remembering that the speed of light is a most extraordinary animal. It may have made sense to Mr Einstein, but I would say that it is next-to-impossible for us mere mortals to get our head around it!

Everything that we (virtually all of us) understand about 'speed' is relative - indeed, the concept of speed has no meaning in the absence of a reference point or observer (i.e. speeds are relative to that point or observer). However,given that we seem to believe Mr Einstein, the speed of light is absolute - i.e. the same 'relative to anything', including 'relative to the speed of an observer who is travelling at high speed (relative to something else).

I don't really know how us 'mere mortals' are meant to conceptualise this, particularly if we think of light as the movement of particles (photons), rather than as 'a wave'. If it were almost any object other than a photon, if it moved X kM away from (or towards) an observer in Y seconds, then its speed relative to that observer would be X/Y kM/sec - so if the observer were moving in the same (or opposite) direction as the object, it's speed relative to the observer, then the speed of the object relative to the observer would be less (or more) than X/Y. How/why on earth that can be different for a photon certainly escapes me!

Earlier in this thread, oops observed that 'mathematics is not a perfect science'. I have to say that as maths and theoretical physics (particularly 'particle physics' and cosmology etc. etc.) gets 'higher' those disciplines seem to converge with what I can but describe as 'philosophy'

 

[JohnW2]

Yup that is the way I initially saw it. Changing from a choice of 1 in 3 to a choice of 1 in 2 (providing that the host - monty hall - always reveals a goat not a car - therefore from .333 to .500 is a 50% increase. ... Simple when you see it but not so simple if you do not.

When you say that's how you 'initially' saw it, are you saying that you are one of those exceptional (but seemingly rare) people who 'immediately saw' the correct (and 'simple') answer?

Another test/survey thingy apparently and goes something along the lines of (others might correct me) :- a disease is reckoned to be 10% prevelant in the community and a test is developed that gives with a 90% probability of a positive result or a negative result.
A patient with no other observations goes to see his GP , he has had that one test and it was positive. What are the chances he has the disease? A/ 90%, B/50%, C/10% , D/5%. Apparently the correct answer is 50% or thereabouts. All thanks to a Reverend Bayes and his Theorem.

I'm not sure what you mean by "90% probability of a positive result or a negative result" (the result must be either positive or negative ) but, yep, as you say, it's all down to Mr Bayes' Theorem, and some people regard it as counter-intuitive. However, I think it's a bit different from Monty Hall.

The important point about diagnostic tests is that the "Positive Predictive Value" (the probability of a person having the disease if the test result is positive) is what matters, and that depends on the prevalence of the disease in the community being tested . If you think about it, that makes some sense, since if, say, the prevalence of the disease is very low, then nearly all positive results (for any test) will be 'false positives'. Ultimately, if the prevalence of disease were zero, then any positive results would be false positives'.

Bayes' Theorem reminds us that, other than for a 'perfect' test, diagnostic tests are not 'absolutes' but, rather arepart of the overall 'melting plot' that helps to refine what we knew before undertaking the test. If the prevalence of a disease is 10%, then the 'pre-test probability' of a random individual having the disease is 10% but the test result then enables us to 'refine' that to a figure closer to the truth!

If you're interested in 'counter-intuitive' matters of probability, are you familiar with "The Birthday Paradox"?

 

[morqthana]

However, it's perhaps worth remembering that the speed of light is a most extraordinary animal. It may have made sense to Mr Einstein, but I would say that it is next-to-impossible for us mere mortals to get our head around it!

Everything that we (virtually all of us) understand about 'speed' is relative - indeed, the concept of speed has no meaning in the absence of a reference point or observer (i.e. speeds are relative to that point or observer). However,given that we seem to believe Mr Einstein, the speed of light is absolute - i.e. the same 'relative to anything', including 'relative to the speed of an observer who is travelling at high speed (relative to something else).

I don't really know how us 'mere mortals' are meant to conceptualise this, particularly if we think of light as the movement of particles (photons), rather than as 'a wave'. If it were almost any object other than a photon, if it moved X kM away from (or towards) an observer in Y seconds, then its speed relative to that observer would be X/Y kM/sec - so if the observer were moving in the same (or opposite) direction as the object, it's speed relative to the observer, then the speed of the object relative to the observer would be less (or more) than X/Y. How/why on earth that can be different for a photon certainly escapes me!

When one asks a "why" question in physics on fundamental postulates, the only real answer is "because" by assuming this, the theory describes the data and it has not been falsified up to now

But hey, maths is not a perfect science,

I think many mathematicians would argue that it is the most perfect of them all.

 

[morqthana]

But hey, maths is not a perfect science, as is evidenced by paradoxes such as Gabriel's Horn. An infinitely large cone which, if filled with paint, would not contain enough paint to cover the outside of the cone.

Gabriel's horn - Wikipedia

en.wikipedia.org

 

[opps]

If you're interested in 'counter-intuitive' matters of probability, are you familiar with "The Birthday Paradox"?

I was chatting to a guy a couple of weeks ago. He recounted how a maths student friend walked into a takeaway and said to the guy behind the counter "I predict that of the next 30 people to walk in, two will have been born on the same day, same month. If I am correct can I have my food for free?". He got his food for free. He knew that he would only have a 50% chance of winning if he said 23 people, so he went for 30.

 

[morqthana]

No downside for him/upside for the takeaway if he was wrong?

Why would the takeaway go for that? Who would ever take a bet where if they lose they pay and if they win they get nothing?

 

[SUNRAY]

No downside for him/upside for the takeaway if he was wrong?

Why would the takeaway go for that? Who would ever take a bet where if they lose they pay and if they win they get nothing?

Doesn't the takaway make a profit on sales?

 

[morqthana]

Ah - you're assuming that if they said "on yer bike" he'd leave and not buy any food.

 

[ebee]

I was chatting to a guy a couple of weeks ago. He recounted how a maths student friend walked into a takeaway and said to the guy behind the counter "I predict that of the next 30 people to walk in, two will have been born on the same day, same month. If I am correct can I have my food for free?". He got his food for free. He knew that he would only have a 50% chance of winning if he said 23 people, so he went for 30.

I once worked it out with a computer program.

OK a Spectrum and I used Spectrum basic.
I used the Random button (OK Pseudo Random, but you get the drift).
I ran it thousands of times and came to the conclusion that 28 days was almost always the number needed to match a birthday with the first entrants birthday at least once.
So I would be inclined to agree that around 29 or 30 people total might be a reasonable figure for a very high expectation of say better than 90 odd percent of the time.

 

[morqthana]

When you say that's how you 'initially' saw it, are you saying that you are one of those exceptional (but seemingly rare) people who 'immediately saw' the correct (and 'simple') answer?

I'm not sure what you mean by "90% probability of a positive result or a negative result" (the result must be either positive or negative ) but, yep, as you say, it's all down to Mr Bayes' Theorem, and some people regard it as counter-intuitive. However, I think it's a bit different from Monty Hall.

The important point about diagnostic tests is that the "Positive Predictive Value" (the probability of a person having the disease if the test result is positive) is what matters, and that depends on the prevalence of the disease in the community being tested . If you think about it, that makes some sense, since if, say, the prevalence of the disease is very low, then nearly all positive results (for any test) will be 'false positives'. Ultimately, if the prevalence of disease were zero, then any positive results would be false positives'.

Bayes' Theorem reminds us that, other than for a 'perfect' test, diagnostic tests are not 'absolutes' but, rather arepart of the overall 'melting plot' that helps to refine what we knew before undertaking the test. If the prevalence of a disease is 10%, then the 'pre-test probability' of a random individual having the disease is 10% but the test result then enables us to 'refine' that to a figure closer to the truth!

If you're interested in 'counter-intuitive' matters of probability, are you familiar with "The Birthday Paradox"?

 

[morqthana]

I once worked it out with a computer program.

OK a Spectrum and I used Spectrum basic.
I used the Random button (OK Pseudo Random, but you get the drift).
I ran it thousands of times and came to the conclusion that 28 days was almost always the number needed to match a birthday with the first entrants birthday at least once.

No, not the first entrant's, just any two people.

So I would be inclined to agree that around 29 or 30 people total might be a reasonable figure for a very high expectation of say better than 90 odd percent of the time.

More, if you want odds that good - 40 gets you to 89.1%.

By the time you get to 100 the probability is 99.99997%

You quickly start needing to use exponential notation, at which point I think it is easier to look at the probability that there are no shared birthdays

All these figures are theoretical, and assume a linear distribution of births per day of year, which is not the case.

 

[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'!

I've just, by chance, read a simple, and IMO, very intuitive explanation.

If you picked correctly the first time, the right move is to stay.

If you picked incorrectly the first time, the right move is to switch.

What were your odds of picking incorrectly the first time? ⅔.

So ⅔ of the time the right move is to switch.

 

[ebee]

When you say that's how you 'initially' saw it, are you saying that you are one of those exceptional (but seemingly rare) people who 'immediately saw' the correct (and 'simple') answer?

sorry John, I missed that one!
Just to clarify, "when I initially saw it" :-
When I looked briefly at the "Monty Hall Conundrum" I viewed it as a simple 1 in 3 random chance in the first instance, then if Monty opens one door to reveal a goat and the choice becomes whether or not to stick with that initial choice or to change to the other unopened door. This becoming a choice of 2.
Therefore from a 1 in 3 becomes a 1 in 2, the chance of being correct increases by 50% only.

When I read that producers of the show that the actual results over many weeks seemed to back up their theory that those who changed their minds appeared to win about twice as much compared to those who did not - a 100% increase whilst I thought it was merely 50% increase.
I think that prof followed similar reasoning to me then changed his mind a few days later to confirm they were correct.
That is the bit I could not see.
After couple of days I thought ho forget it.
Two weeks or more later, never having consciously considered it any more, I was having a shave one morning and half way thru that shave I had a Eureka Moment, it suddenly twigged.
a 1 in 3 chance of success and a 2 in 3 chance of failure gets turned upside down (or back to front really) if you change your mind .
2 in 3 chance of success and 1 in 3 chance of failure - simple once the penny drops as folk says.
.