The Monty Hall problem is this...
You are offered a choice between three doors; two conceal garbage and the 3rd conceals £1m.
After you have chosen a door, another door is opened alwyas revealing garbage.
You are then offered the opportunity to swap or retain your door in a situation where there are two doors in play, one concealing garbage and one concealing £1m.
Should you swap?
The answer is yes, always.
You can go through complicated maths and probabilities or you can reduce it to simply this;
If you stay with your original choice, you’ll win only when it was correct; which happens only 1 in 3 times. If you swap, you’ll win whenever your original choice was wrong, which is 2 out of 3 times.
Now the question is "Does this apply to the swap offered at the end of DoND?". For which there seems to be a great deal of argument;
You choose a box out of 22 (I think). 20 are then opened leaving two boxes, the one you have and the unchosen one. You know the two amounts involved, but you have no idea which is which.
Should you switch? Or does it make no statistical difference?
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Deal or no deal? How passé :-)
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Never mind the choice... you're watching DoND... and missing Flog It?
Last edited by: Fenlander on Tue 24 Sep 13 at 15:00
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>> Never mind the choice... you're watching DoND... and missing Flog It?
You fancy yourself as a bit of a divvy tho, don't you FL.. Perhaps you need to rename to Lovejoy?
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Flog it ? Only worth watching when Catherine Southon is on.....she could flog me !.... Phoooar !
Ted
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I thought I was the only one - there's something about that lass!
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>> I thought I was the only one - there's something about that lass!
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Hands off....I saw her first !
Ted
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>> >> I thought I was the only one - there's something about that lass!
>> >>
>>
>> Hands off....I saw her first !
>>
>> Ted
I'm really in love with Carol Kirkwood!
Mind you I only tune into BBC Breakfast to look at the ladies legs, particularly Louise Minchin's long 'uns, and the one day last week I missed looking in, Susanna Reid did a panty flash!
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I guess you missed Rachel Riley on a Spacehopper on "Eight out of Ten Cats does Countdown" I guess ;-)
Here's one I haven't seen before...
www.youtube.com/watch?v=UH67RGKQBFk
Last edited by: Lygonos on Tue 24 Sep 13 at 22:48
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Hopeless video. Stopped watching after thirty seconds because they cut it, and I didn't have time to work out a word from the letters or see how the contestants did.
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>>Stopped watching after thirty seconds because they cut it, and I didn't have time to work out a word
Go back and check 1:00 then - appropriate wordage.
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>> Should you switch? Or does it make no statistical difference?
There are factors at play here, notably you have chosen, or were give the box right at the start.
That is your fate and should be stuck with, unless of course its clear from the logical evidence laid in front of you that your hand is crap.
So the answer is in your example, you stick with it.
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This is a famous problem and the answer is that you should switch.
The difficulty is in explaining that.
You don't actually know of course whether you have picked the right one or not. The host must, because he has to remove one that doesn't contain the prize.
Does that make any difference to the odds? If you stick, no. Because the odds were 1/3 when you picked one of three boxes. How could the odds change, there is always going to be one empty box for the host to remove?
If you switch, yes. There are now two boxes, one of which contains the prize. Choose the other and your odds must be 2/3. The reason is that if the odds were 1/3 with your box, the balance of the odds which must total 1 are with the only other remaining box.
No I'm not intuitively convinced either, but there are simulators on the web so you can test the theory.
Another way to look at this is to pretend that the host is removing all the boxes that you didn't pick except one. Say there were 1000 boxes and only one prize. You choose one. The host removes 998 and says "do you want the one you picked, or this one here?" It's obvious that it's very unlikely that you picked the right one out of 1000, so you switch, right?
I think I first came across this in Martin Gardner's 'Mathematical Puzzles and Diversions'.
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>> This is a famous problem and the answer is that you should switch.
Are you talking about the DoND problem though?
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Monty Hall. I have no knowledge of Deal or No Deal.
Can the principle be applied?
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Not really because all the boxes have something, and you know whats in the other boxes as the show goes on, but not by individual box. The odds keep changing as boxes are eliminated.
And you can sell your box before you get to the choice.
Last edited by: Zero on Tue 24 Sep 13 at 15:46
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Re-reading No FM2R's explanation, I don't think we know enough. But, surmising that there is a really good prize and a crap one, and that there was only one really good prize, the Monty Hall answer would apply.
It would be more akin to my hypothetical example of 1000 boxes. Your odds would go from 1/22 to 21/22.
If there were two good prizes, your odd would go from 2/22 to 20/22, assuming the host always leaves a good prize if you haven't picked one. And so on.
If the host is choosing randomly, it's a different problem.
Do we know how many categories of prize there are and what the distribution is? My instinct (a very bad guide to probability problems usually) is that if the prizes are mostly crap, but if you know that one of the boxes you are choosing from contains a good one, you should switch.
EDIT: Sorry, crossed with you Z.
Is it worth watching?
Last edited by: Manatee on Tue 24 Sep 13 at 15:57
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You can play around with the maths all you want, but it's easy to see switching could not make the slightest difference to your chances of opening the correct.
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>> You can play around with the maths all you want, but it's easy to see
>> switching could not make the slightest difference to your chances of opening the correct.
Why? Do I have to watch it to find out?
Is there always a top prize and a rubbish one in the final pair, and is the one "owned" by the contestant randomly picked? If so, my instinct would be to swap as it would be akin to the Monty Hall problem. But as I haven't a clue how it works ...
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If it helps, no one knows what's in the closed boxes, apart from by the process of elimination as they are opened.
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>> >> You can play around with the maths all you want, but it's easy to
>> see
>> >> switching could not make the slightest difference to your chances of opening the correct.
>>
>> Why? Do I have to watch it to find out?
>>
>> Is there always a top prize and a rubbish one in the final pair, and
>> is the one "owned" by the contestant randomly picked? If so, my instinct would be
>> to swap as it would be akin to the Monty Hall problem. But as I
>> haven't a clue how it works ...
No you could end up with a a super prize and a not quite as super prize as your final two boxes. So you then have to choose between 100k or 250k. And earlier on the banker may have offered you 125k to sell your box that may contain 100k or 250k
you need to watch it to get the idea. Its akin to poker.
Last edited by: Zero on Tue 24 Sep 13 at 17:09
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Manatee - at the risk of stating the obvious, the banker doesn't know what's in the boxes either (no one does)
Last edited by: Focusless on Tue 24 Sep 13 at 17:11
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>> Manatee - at the risk of stating the obvious, the banker doesn't know what's in
>> the boxes either (no one does)
Not obvious to me - I might have to watch it! It does sound less obvious, if that's the right word, than the Monty Hall version.
Does being akin to poker mean that everyone knows what's in the pack?
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DoND is rubbish.
The banker is not trying to win, and is not offering amounts solely driven by what is available.
The competitor is usually, shall we say, misguided but trying to be a hero, whereas everybody else is there to make a television show which offers long term appeal to its target audience and is quite happy to stump up a prize from time to time to do so.
Like him or not, and I do like him, Noel Edmonds is a genius at that type of entertainment/manipulation.
If you played scientifically and logically you would be boring to that audience and thus unwelcome and unlikely to pass through the selection stages.
Which doesn't impact my interest in the many factors at play.
Last edited by: VxFan on Mon 14 Oct 13 at 00:57
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>> Manatee - at the risk of stating the obvious, the banker doesn't know what's in
>> the boxes either (no one does)
>>
Yeah... right!
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>> EDIT: Sorry, crossed with you Z.
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>> Is it worth watching?
It was a first yes. There was an element of real drama, angst, pathos, despair, anger from the victim at times. For example selling their box that contained £250K pounds for £9k pounds (you have to play the game through even tho you sold - as along the classic line of Jim Bowen (Bully lets see what you might have won))
But now, its been "themed" and it has catchphrases its now a LooB. But you need to watch it once or twice just to get an idea of the gambling and logic involved.
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I never even look at stuff like that, big brother, come dancing, there are endless rubbishy game shows... cooking, my God, with a special cup for backbiting (I have seen one or two of those).
Childish pap for the goggling moron.
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You all need to turn the TV off and get outside for some fresh air...there will be so many days ahead of us where it won't be possible!
Lorry driver speak= Get a life!
Pat
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It's the Autumn TV season started. Cop-show heaven! Stana Katic in "Castle" - whoar!
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>>>You fancy yourself as a bit of a divvy tho, don't you FL.. Perhaps you need to rename to Lovejoy?
Actually I know a divvy as an idiot or waster... I'm sure you didn't mean that so you will have to expand on what you think I fancy myself as.
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>> >>>You fancy yourself as a bit of a divvy tho, don't you FL.. Perhaps you
>> need to rename to Lovejoy?
>>
>> Actually I know a divvy as an idiot or waster... I'm sure you didn't mean
>> that so you will have to expand on what you think I fancy myself as.
what's known as a 'divvy' (diviner) in the antique business, can more or less smell a genuine antique in a room full of rubbish.
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Going back to the subject the problem caused me quite some head scratching and the standard of 'explanations' on the web seems rather lacking, to put it politely.
So I pondered, and came up what I think is both correct and reasonably understandable. Not on my index page yet as not polished but you are welcome to read the beta version.
www.slidingpillar.co.uk/monty_hall.htm
Not got round to drawing doors, goats, or a Cadillac yet...
There is no need to consider 'filters' by the way, the real point is there are multiples of three games, not two.
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There are two different choice points.
When you made your original choice there were three possibilities, one correct one and there is a 1/3 chance that you have chosen correctly.
Then a single wrong answer, which you did not choose, is removed. and you are offered another chance to choose.
This is now a different game and there are now only a subset of the original possibilities available.
It matters not what the odds were, it matters what the odds are. Not only that, what the odds were has no bearing on what the odds now are.
And the odds are now that there is a 1/2 chance that you originally chose correctly. And if you change, there is a 1/2 chance that you will be wrong to do so.
It is not relevant to your odds of winning whether you change or not.
Last edited by: No FM2R on Tue 24 Sep 13 at 18:40
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It is not unlike the other problem of three people giving you £10, you return £1 to each of them etc. etc. etc.
It encourages you to do the wrong sum.
I think.
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>> Going back to the subject the problem caused me quite some head scratching and the
>> standard of 'explanations' on the web seems rather lacking, to put it politely.
>>
>> So I pondered, and came up what I think is both correct and reasonably understandable.
>> Not on my index page yet as not polished but you are welcome to read
>> the beta version.
>> www.slidingpillar.co.uk/monty_hall.htm
>> Not got round to drawing doors, goats, or a Cadillac yet...
>>
>> There is no need to consider 'filters' by the way, the real point is there
>> are multiples of three games, not two.
>>
I think a simpler explanation is...
On your first pick, you clearly have a 1/3 chance of being correct.
There is a 2/3 chance that the Cadillac is in one of the other two boxes.
The host removes a 'goat' box from the two you didn't pick.
There is still the 2/3 probability that the Cadillac was in one of those two boxes...and you know which one it isn't in. So you only have to pick the other to have that 2/3 chance.
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>>There is still the 2/3 probability that the Cadillac was in one of those two boxes
I don't think so. The problem is the word "still".
The second time you are asked a question you are facing a different situation.
Because a box has been removed there is now a 50% chance that you originally chose correctly, not a 33% chance.
Thus, you are now playing with a 50:50 solution.
Last edited by: No FM2R on Tue 24 Sep 13 at 19:09
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Thus, you are now playing with a 50:50 solution.
Nope.
Best way to visualise is run all possible games.
It reduces down to three games, but you are welcome to consider more, but you'll find the answers divisible down to a basic three with a 1/3 chance if you stick and 2/3 if you switch.
You can also run a computer program to simulate lots of games and get the same result.
Basically the Cadillac can be behind doors 1, 2 or 3. You can pick any of them but if you don't pick the door with the car, Monty is going to improve the odds for the switch door as he knows where the car is, and is always going reveal a goat.
This is where the intuitive point about it being a simple two door problem breaks down as put quite simply, it is not that simple; the odds are altered by the fact that Monty knows what is behind the doors.
Schroedingers cat here kitty, kitty does have some similarities here...
Last edited by: Slidingpillar on Tue 24 Sep 13 at 19:37
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>>Thus, you are now playing with a 50:50 solution
Clearly not, since it's easy to simulate and 2/3 of the time switching will get you the winning box, on average.
You can't avoid the fact that you had a 1/3 chance when you made your first choice. The removal of a goat box by Monty doesn't change that or give you any new information about 'your' box.
Given the 1/3 odds for your box, the other box- now the only option - must have 2/3 probability.
Put another way, there's remains the 2/3 chance you were wrong the first time and the prize was ion one of the other two. You now only need to pick one box to have that 2/3 chance.
www.grand-illusions.com/simulator/montysim.htm
(told me it might not work in my browser, Chrome, but it does)
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>>there's remains the 2/3 chance you were wrong the first time
When you made the choice that was true. However, surely now there is only a 1/2 chance that you were wrong the first time.
Last edited by: No FM2R on Tue 24 Sep 13 at 19:41
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Just sent to me (seems difficult to fault) ...
"Instead of one Monty Hall experiment, you do 3000 experiments at the same time. A large arena is rented, 3000 tables are set up, and people are hired to place one red card and two black cards on each table. At each table, the worker places the red card according to a die roll, so that the position is random (with uniform probability). All three cards are face down.
After the above is done, you enter the arena with 3000 pebbles. You go to each table and place one pebble on one card.
Almost all the time, about 1000 of your pebbles will be on red cards and about 2000 will be on black cards, by random chance.
Now I go to each table, examine the cards, and turn over a card that is black and does not have the pebble on it.
How many pebbles are on red cards? It MUST be the same number as before, because the pebbles have not moved and the cards under them have not changed.
At this point, we have about 1000 pebbles on red cards and about 2000 pebbles on black cards.
If, at each table, you switch the pebble from the original card to the only other card that is still face-down, then we have about 2000 pebbles on black cards and about 1000 on red cards."
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>>At this point, we have about 1000 pebbles on red cards and about 2000 pebbles on black cards.
>> If, at each table, you switch the pebble from the original card to the only other card that is still face-down, then we have about 2000 pebbles on black cards and about 1000 on red cards.
3000 tables, 1000 have a pebble on a red, and 2000 have a pebble on a black.
EVERY table (3000 in total) has a red and a black, therefore when you switch them over you have 2000 red and 1000 black.
What you have linked is simply wrong in the last line.
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>>What you have linked is simply wrong in the last line.
Indeed. Thanks. I offered the correction but didn't mention that it wasn't actually me who spotted it.
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>> >>there's remains the 2/3 chance you were wrong the first time
>>
>> When you made the choice that was true. However, surely now there is only a
>> 1/2 chance that you were wrong the first time.
Don't think so.
Imagine 100 people did this simultaneously, and they all 'stuck'. Only 33 of them, more or less, would win the Cadillac because only 1 in 3 would have picked the right one in the first place. That would mean there were 67 Cadillacs, more or less, in the other 100 boxes.
We already know that the Monte Carlo simulation will prove this - all we are trying to do is explain it :)
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>>all we are trying to do is explain it
I know, bloomin annoying.
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>>>what's known as a 'divvy' (diviner) in the antique business, can more or less smell a genuine antique in a room full of rubbish.
Ahh OK... yes I can do that... been involved on the sidelines for decades and had my "training" in the 70s when there was a lot of good stuff about. What passes for antique these days is mostly quite modern and stuff we'd have made into firewood back then.
Wasn't the expression created by the Lovejoy program and one not actually used by the trade?
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Hence the reference to Lovejoy in my original post..
Perhaps your nose is not that good after all.......
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>>>Hence the reference to Lovejoy in my original post.. Perhaps your nose is not that good after all.......
Not for fictional TV scripts obviously... the real life stuff no problem.
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It must be a regional thing Fenlander, because that's my interpretation too!
Pat
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Apologies if my skimming of this thread missed someone else making this point, but DoND is not a Monty Hall problem because nobody knows what is in any of the boxes, so no-one is in a position to offer the contestant clues based on that knowledge.
Monty Hall, on the other hand, knew which was the 'good' box, and could give away a little of that information by taking away a box he knew to be 'bad'. The contestant could then use that information to reassess his original choice of box. Without that extra information, the DoND contestant can only add up the unrevealed prizes, divide by the number of boxes remaining, and decide whether the bank's offer represents good value against the risk of receiving significantly less. I doubt many manage even that.
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DoND rules here, although they don't mention that the boxes contain amounts between 1p and £250,000 (vaguely logarithmic scale):
www.dealornodeal.co.uk/show/game-rules/
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Isn't there a philosophical point here - can past events alter the chances of a new event occuring?
All the making of choices in round 1 is just a charade. The fact is at round 2 there are two choices, one wins, the other doesn't. You can't look at one of the options and say that its past history determines its chance of being one or the other, because whatever chance it has will be mirrored by the other.
Imagine someone else had played the first round at the same time, and was now sitting in the other chair. It would be ridiculous to say they each improved their chances of winning by swapping seats.
It's like throwing 5 sixes in a row and saying there is a bigger chance of throwing another.
Or surviving Russian roulette twice and deciding to have another go.
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You're right, Cliff, but the "past events have an influence on what is about to happen" view becomes so seductive that you end up with, for example, the Challenger disaster, where according to Feynman there was a lot of "it's always been fine so not only will it be fine this time, the odds are actually improving that it will be fine" type of thinking.
Monty Hall variants, however, are pretty easy to suss, and obviously it's better to swap. For some reason some people can't appear to work it out, or over-think it, and hence Mr Monty makes his easy money, and always has done.
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I'd rather open the box than take the money: www.youtube.com/watch?v=irg29je8G8k
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Crikey Dog - I'm reminded of a tv advert in the late sixties, I don't know for what, that had the slogan "Go on - take the box", at which the family would collectively cry "No, open the money".
I think families have a lot to answer for.
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>> I'm reminded of a tv advert in the late sixties, I don't know for what, that had the slogan "Go on - take the box", at which the family would collectively cry "No, open the money".
I must have missed that one Cc, never missed Take Your Pick on a Friday evening though :)
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The Monty Hall Problem and Deal or No Deal - Roger.
