Estimating the age of the Universe, and getting it all wrong

[Jon]

Now Monty opens a door and shows a zonk. The probabilities MUST still add up to 1 because there IS a car back there somewhere, so the higher math types say that all of the probability that vanished from the revealed door must have gone to the other doors. It is here that mathematicians do it wrong.

The door you picked has NOT been revealed yet. One other door hasn't been revealed yet. But if you say that the unpicked door probability just became 2/3, you did something wrong because you added all of the probability from the revealed door to one door - without justifying this action. You SHOULD have added 1/2 of the original probability of the "reveal" door to each of the still-closed doors, or 1/6 to the picked door and 1/6 to the remaining door. If you do, their probabilities should be 1/2 each.

Ok Doc, I understand your argument now. So I have a simple question for you...

=> If you believe the odds they state are incorrect, how do you account for the fact that an empirical test confirms their odds to be true and disproves your own hypothesis?

 

[The_Doc_Man]

@Jon,

Because the answer to this question depends on which game you are playing. More specifically, which RULES you are applying. I stated the rules I was using. I developed a spreadsheet to completely cover the actions at each stage of the game. I admit I did a short-cut, but symmetry says that I would get the same exact results if I put the car behind door #2 or door #3 and ran those probabilities again. I came up with 50/50 odds using the rules as stated. I cannot account for the fact that someone erred in their analysis but I can tell you where they probably erred.

After the "pick" and "reveal" steps, Monty shows you a zonk. At that point, you have a picked (but as yet unrevealed) door and a remaining unrevealed door. People say that at this stage, the probability of the prize being behind the remaining door is higher than the odds of it being behind the picked door. This is because they assign ALL of the probability from the "revealed" door to the "unrevealed AND unpicked" door. But there is no justification for this asymmetric assignment to that third door. Since neither the picked nor the unpicked door have been revealed, there is no reason to assign probabilities that way. You must assign the excess probability EQUALLY to both doors.

There is another place where there is (I believe) an improper mapping of options. Once you pick a door and it happens to be the right one, Monty will STILL give you a chance to switch (according to the rules I stated). But now he has two doors he could reveal, either of which is a zonk. If you DON'T show this as two different scenarios, then you get the odds of 1/3 stay, 2/3 switch. This is the only place where Monty actually has a choice, as the rules would preclude one of the "reveals" for the case where you DIDN'T pick correctly. In those other cases, Monty's choice is constrained to one and only one door that can be revealed.

If you do a web search for the "Monty Hall" problem you will see literally hundreds of articles on the subject. The odds are 50/50 as to whether you will find articles claiming better odds or equal odds of switching. ( ) But all of the articles clearly claim

 

[Jon]

@Jon,

Because the answer to this question depends on which game you are playing. More specifically, which RULES you are applying.

We are playing the same identical game as outlined in the video. The rules are identical. Given that, how is it that the empirical results disprove your hypothesis?

This is because they assign ALL of the probability from the "revealed" door to the "unrevealed AND unpicked" door. But there is no justification for this asymmetric assignment to that third door.

Why do you believe there is no justification? They explain the justification very clearly. At the start of the game, the total probability of the car being behind doors 2 and 3 as an aggregate is 2/3 rds. When you find that door 2 does not have the car behind it, the aggregate of probabilities for that group (2 and 3) is still 2/3 rds. But since you are certain it is not behind door 2, the odds are it is 2/3 rds behind door 3.

There is endless debate. But here's the rub. Many of those you cite who do not want to do the extremely simple experiment are just protecting their incorrect hypothesis, and would rather defend a mistake than realise they are in fact mistaken. Probability is a rather odd creature and not always intuitive. But people assume they understand it, despite not studying it. And that leads to mistakes. Heck, I made constant mistakes when I studied it and still do.

In science, you test hypotheses. Untested hypotheses are theory, tested ones that are confirmed are facts.

Or, to boil it all down...

=> Have you tested your hypothesis? I have.

Edit: Maths with statistics and probability at A Level used to do my head in. Then again, I was a bit of a late developer.

 

[Jon]

I found this table that explains the potential outcomes very well:

Source: https://en.wikipedia.org/wiki/Monty_Hall_problem

In fact, the Wikipedia article is very good. @The_Doc_Man You are in good company, with nearly 10,000 readers of Parade saying the switching strategy was wrong. But here is the thing that tells me about human nature...

** Even after showing simulations, they were still unconvinced! ** i.e. when reality confronts hypothesis, people get stuck in their belief and refuse to budge.

Source is from this quote from the Wikipedia article:

Many readers of vos Savant's column refused to believe switching is beneficial and rejected her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling vos Savant wrong.[4] Even when given explanations, simulations, and formal mathematical proofs, many people still did not accept that switching is the best strategy.[5] Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating vos Savant's predicted result.[6]

I find it a bit odd though that all these people haven't tested it.

And here is something interesting about this. Pigeons learn to switch, while humans don't! OMG!

Source:

In his book The Power of Logical Thinking,[21] cognitive psychologist Massimo Piattelli Palmarini [it] writes: "No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer." Pigeons repeatedly exposed to the problem show that they rapidly learn to always switch, unlike humans.[22]

 

[Jon]

Does anyone have an example of another paradox that melts the brain?

 

[JonXL]

If someone asked you, "What is the life expectancy of a man in the UK?", would you choose the average age of death (or even the median (midpoint)), or the age with the highest frequency? I think the answer is obvious.

I don't know that the answer is obvious at all. Our discussion should show that a question like that can have different answers depending on the aim. So a sensible response might be "what man?". That then leaves the clarification of an average man or most men (among other possibilities). That question can then be answered, but the answer won't be the same for an average man and most men - as we've seen.

The principle, as it relates to guesstimating the age of the Universe, is that choosing the midpoint will give you a statistically more accurate chance of being closer to the truth than choosing a non-midpoint. Did that make sense?

The applicability of this to guessing the age of the Universe is hard to see. To make the problem of ages of people in the UK as we've been using them applicable to the issue of "how old is the Universe", we'd have to have data on millions+ of universes most of which would have had to have been born and died within our window of observation. Then we could ask both our questions (average and mode) about our own universe - assuming we had data on so many others but not our own - and get answers for each.

But that isn't how scientists do that for figuring out the age of the Universe - they don't start from a general value of universe longevity observed in millions of other universes and then try to guess where that would place our particular universe. Instead, they start by actually figuring out the age of the Universe based on observations of things within this universe. They then guess at what the likely lifespan of this universe will be based on how old it currently is and some other principles (like the probability that we are living toward the beginning or end of its lifespan specifically vs somewhere in the middle of its lifespan).

In other words, scientists start with "how old is the Universe" and then estimate how long it will live. You seem to want to start with how long the Universe will live and then guess at how old it currently is. That, I think, is backwards from how it's really done.

 

[Jon]

I don't know that the answer is obvious at all. Our discussion should show that a question like that can have different answers depending on the aim. So a sensible response might be "what man?". That then leaves the clarification of an average man or most men (among other possibilities). That question can then be answered, but the answer won't be the same for an average man and most men - as we've seen.

I think you are picking hairs here and ignoring common vernacular. Furthermore, considering we are talking about the context of guesstimating the age of something knowing nothing else about them, it is obvious to most people they are talking about the average.

The applicability of this to guessing the age of the Universe is hard to see. To make the problem of ages of people in the UK as we've been using them applicable to the issue of "how old is the Universe", we'd have to have data on millions+ of universes most of which would have had to have been born and died within our window of observation. Then we could ask both our questions (average and mode) about our own universe - assuming we had data on so many others but not our own - and get answers for each.

You do not need millions of universes to sample estimated age. You are conflating "taking the principle of using a midpoint to get the best guess" with "taking a sample of a group, and getting the average from that." It is the midpoint principle that is relevant to guessing the age of the Universe. The talk about groups of people and averages is just to give an example that illustrates how you would minimise the error demonstrably by taking a midpoint, or average in the case of a group.

The best guesstimate to minimise error in guessing the age of ONE PERSON is to choose the midpoint. You agreed this yourself. In other words, taking the halfway position is a good way to estimate the age of something to have the best chance of being closest to the truth. I believe it follows from that. Do you agree?

But that isn't how scientists do that for figuring out the age of the Universe - they don't start from a general value of universe longevity observed in millions of other universes and then try to guess where that would place our particular universe.

Again, as I mentioned earlier, the multiple Universe requirement is unnecessary. The principle is the midpoint. You don't need more than one Universe to use this principle.

Instead, they start by actually figuring out the age of the Universe based on observations of things within this universe. They then guess at what the likely lifespan of this universe will be based on how old it currently is and some other principles (like the probability that we are living toward the beginning or end of its lifespan specifically vs somewhere in the middle of its lifespan).

And what if their hypothesis is statistically impossible? If the Universe does not have an end in time, then the midway point is infinity. Consequently, the probability of these scientists being correct by stating the Universe is 13.8 billion years old is infinitely small. i.e. zero chance.

In other words, scientists start with "how old is the Universe" and then estimate how long it will live. You seem to want to start with how long the Universe will live and then guess at how old it currently is. That, I think, is backwards from how it's really done.

How can you start with "how old is the Universe" by observation if by logic it is statistically impossible? An interesting thought experiment, me thinks.

 

[The_Doc_Man]

I found this table that explains the potential outcomes very well:

View attachment 97285

Source: https://en.wikipedia.org/wiki/Monty_Hall_problem

In fact, the Wikipedia article is very good. @The_Doc_Man You are in good company, with nearly 10,000 readers of Parade saying the switching strategy was wrong. But here is the thing that tells me about human nature...

** Even after showing simulations, they were still unconvinced! ** i.e. when reality confronts hypothesis, people get stuck in their belief and refuse to budge.

Source is from this quote from the Wikipedia article:



I find it a bit odd though that all these people haven't tested it.

And here is something interesting about this. Pigeons learn to switch, while humans don't! OMG!

Source:

If you look at the table I put in my post, you will see that I went deeper. There is a problem here. The "Result if switching to door offered" glosses over some chances, whereas my offering does not. This offering does not correctly count all possible options.

 

[The_Doc_Man]

As to your "pick the age" problem, the correct answer from statistics is to always pick the mean age as opposed to the median or mode ages. This gives you the greatest probability of being closest to the correct age most of the time.

 

[Jon]

As to your "pick the age" problem, the correct answer from statistics is to always pick the mean age as opposed to the median or mode ages. This gives you the greatest probability of being closest to the correct age most of the time.

I agree Doc with using the mean for a group. The problem with using a mean when guessing the age of the Universe is that you have no group to average. So a fudge would be to choose the halfway point. Using this principle, you can best estimate its potential age. The unknowns are when it started and when it will finish, but you can look at the scenarios conceptually.

Scenario 1: Infinitely old. Halfway point = infinity

Scenario 2: 100 trillion years old. Halfway point = 50 trillion years

In Scenario 1, if the halfway point is infinity, the likelihood of the scientists 13.8 billion years old estimate being close is virtually zero.

In Scenario 2, the scientists prediction is likely to be way off, and therefore very probably false.

 

[conception_native_0123]

when it will finish

there's all types of theories for that Jon. the big freeze, the big rip, and so on...

 

[The_Doc_Man]

I agree Doc with using the mean for a group. The problem with using a mean when guessing the age of the Universe is that you have no group to average. So a fudge would be to choose the halfway point. Using this principle, you can best estimate its potential age. The unknowns are when it started and when it will finish, but you can look at the scenarios conceptually.

Scenario 1: Infinitely old. Halfway point = infinity

Scenario 2: 100 trillion years old. Halfway point = 50 trillion years

In Scenario 1, if the halfway point is infinity, the likelihood of the scientists 13.8 billion years old estimate being close is virtually zero.

In Scenario 2, the scientists prediction is likely to be way off, and therefore very probably false.

But you see, the estimate of the age of the universe isn't based on guesses. The measurements of movement of the stars and galaxies is away from us in every direction (which tends to support the expanding universe hypothesis). But math allows us to look at the rate of expansion and reverse it all (virtually, of course) to see that point when everything was at a common point - the singularity. Your comment in scenario 1 is therefore ignoring some fairly good science regarding red-shift and other computations. There is no guess unless you want to refer to a very well-educated guess.

 

[Jon]

They thought the rate of the Universes expansion was slowing down, but they found out that in fact it was speeding up. It was counter-intuitive and against common belief and assumption. Science can get things back to front.

Using observation is one paradigm through which to view something. It relies on models, and assumptions of what you can see. You can look at a piece of paper with pie written down to infinity, yet you cannot see that infinity as it is not observable. Yet conceptually you know it to be the case.

You cannot see gravity yet Newton created the equation. You can only see the effects of gravity. But you can prove the unseeable conceptionally. I'm entering waffle territory here, but I am enjoying it.

The problem as I see it with assumptions about the Universe and its age is that it relies on untestable theories. We don't know if the Universe was born out of nothing or just a re-explosion following an earlier contraction. Or could there be something from a different dimension that intervened and since we are limited to our 3 dimensions and time, the plane of the other dimensions are unavailable to us. They are not observable, although some might argue they can be implied. I've read some stuff about bubble multi-verses where a membrane from one collides with another Universe and it causes some effect. We see the consequences, but not the cause.

It is all a very interesting topic. The James Webb telescope will increase our observable scope and understanding. Who knows, it may profoundly change the current assumptions on how everything began.

 

[Jon]

there's all types of theories for that Jon. the big freeze, the big rip, and so on...

There is another theory: the big unknown. The problem is that most people think too much.

Edit: Only kidding. It is a subtle joke between me and Mr Mysterious.

 

[Steve R.]

There is another theory: the big unknown. The problem is that most people think too much.

My concern is that we are attempting to analyze something (age of the universe) based on what is perhaps a faulty model (big bang theory).

 

[The_Doc_Man]

There are details about the "big bang" that are indeed faulty, but there are now multiple separate approaches that agree the universe had a distinct starting point and the approximate ages are converging in the 12-14 billion year range.

There are essentially three ways to compute the age.

1. Math regarding the Hubble and other constants.
2. Measuring the rate of expansion and then extrapolating backwards to the singularity
3. Recognizing that the universe CANNOT be younger than the oldest objects therein. (Globular clusters play a part in this.)

WMAP- Age of the Universe

Public access site for The Wilkinson Microwave Anisotropy Probe and associated information about cosmology.

wmap.gsfc.nasa.gov

Age of the universe - Wikipedia

en.wikipedia.org

How Old is the Universe?

The age of the universe is determined by the expansion rate of the cosmos and the standard model of cosmology.

www.space.com

 

[Minty]

Casinos work on guesses and the house nearly always wins. They guess they will come out ahead. But if they are just guessing, how come they nearly always do come out ahead? => Probabilities and statistics.

Not true about them guessing - pure probabilities being weighed in favour of the house in almost every game in a casino.
Roulette is the obvious one - House odds are 35-1 and downwards, but there are 37 or 38 numbers on the wheel with the 0 and/or 00.

EUROPEAN​		AMERICAN​
NO	DOUBLE ZERO	YES
37	NUMBER OF SECTORS	38
2.7%	HOUSE EDGE	5.26%
97.3%	AVERAGE PAYOUT FOR 100 €	94.74%

So a guaranteed bias in favour of the house. Others are similar.

 

[RogerCooper]

Not all times in the history of the universe have an equal probability of intelligent life asking questions. The evolution of intelligent life requires stars of a certain size and stability. Ten Billion years from now, such stars will not exist.

 

[Jon]

Not true about them guessing - pure probabilities being weighed in favour of the house in almost every game in a casino.

You seem to have misinterpreted my paragraph. If you are playing Blackjack, for example, the house will employ certain rules on whether to stick, twist etc. They are guesses. The casino cannot be certain whether their guess was correct and will win the hand. But over time, these guesses and their associated probabilities accumulate to put the odds in the casinos favour.

If you think guess is the wrong word to use, why not look at Google's definition?

estimate or conclude (something) without sufficient information to be sure of being correct.

Are you suggesting that Casinos do have sufficient information to be sure of being correct? Since they work on probabilities, they can be never sure, only likely (and over the longer term).

And of course casinos work on probabilities that work in their favour. I thought everybody knew that. However, I do have a friend who lost nearly his entire wealth gambling down the casino. Trying to rationalise with him that he is fighting an uphill struggle trying to beat the house was a fruitless task. Nowadays, he has put most of this money into bitcoin. This is a far better option than the casino, although still high risk.

 

[Minty]

I'm still not sure that guess is the correct choice of word.
Blackjack is played by the dealer to very specific rules as to when he twists or hits, they don't deviate from these rules as they are calculated by the odds to ensure he plays to win in the houses favour. Casinos also add extra decks to move the odds more in their favour.

From that perspective there is no guessing, it's a calculated risk, with the odds definitely slightly in their favour, which is all they need.
A (highly) calculated risk based on the probability of having studied thousands of hands and done lots of maths, just doesn't feel to me like a guess. Semantics. They are very sure of the probabilities and the net outcome.