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peterbirksWell, what a pleasant evening. I had a coffee at a beachside cafe – ridiculously expensive if you look at it in one way, and outstandingly cheap if you look at in another. Because it’s the little pleasures in life that count, isn’t it? And this was just such one of those. Peace, harmony, balance.
Another sunset on the beach
The promenade. I liked the generation of shadows.
I thought that this picture would come out slightly better than it did. Perhaps it needs some rebalancing of the colours.
Woohoo, segways! The police also use these (as well as bicycles) in the pedestrianised/tramway/bicycle areas of the town.
Place Massena by night.
The western end of Rue de France.
And then I went for a walk with the intention of going into the Vielle Ville. However I was sidetracked by a brasserie – a PMU during the day! – that offered a kilo of moules mariniere and chips for E10.90. Topped off with a fromage blanc with raspberry juice on top (a little too sweet for my tastes, perhaps) and a coffee. And, well, a kilo of moules mariniere! Yumbo yumbo.
On top of that, I discovered that my Citibank card, which I assumed would be useless here because there are no Citibank branches in France, actually works at all BNP Paribas branches. No idea what exchange rate I shall be charged, but it’s a neat way to turn dollars into Euros.
Anyhoo, while pondering these unanswerable maths questions, I thought about an interesting “well-known” probability case that I suddenly realized was flawed.
I spotted the flaw because this is what I do all the time when thinking about poker. What intrinsic assumptions do people make when they play the game? By this I mean, when people say that “Position is an advantage”, what assumptions are made (implicitly) for this to be true? Are there in fact no assumptions? Is position always an advantage? (The answer to that last question, btw, is “no”. From that you can work out why position is usually an advantage. Unfortunately I haven’t yet been able to crack the almighty question of “in monetary terms, how much is position an advantage, given other, known, factors?” But I’m getting there).
People make assumptions in probability all the time. One of my favourites is the “Do you change your mind?” problem. I like it because it catches out just those people who know a bit, but not quite enough.
You phrase the question thus:
“You are on a quiz show. There are three doors. Behind one is a goat. Behind another is a car, and behind the third is nothing. You need to beat three other contestants to get to this point. You beat the other contestants. The quizmaster asks you to pick a door. You pick door C. The quizmaster then opens door A to reveal the goat. Assuming that you want to win the car, do you change your mind?”
The mathematical novice says “it doesn’t matter. You had a one-third chance of being right. Nothing has happened to change that.”
The guy who thinks he’s a bit cleverer than that says to the mathematical novice: “Ah, you donk! Of course you change your mind. By showing you the door that contains the goat, the quizmaster had to make a choice. Therefore it’s twice as likely that the other door contains the car than the door you first chose.”
At which point the man with the plan (let’s call him Jeff Duval, for want of a better choice) says: “Donktastic. No, you do not change your mind. Have you never watched this programme before? The quizmaster only opens one of the doors to reveal a goat or a non-win if you have made the correct choice in the first place. If you have made the wrong choice, he just says “Are you sure? Do you want to change your mind?”
The point here is that implicit in the choice of the guy who thinks he is a bit cleverer than he is is the assumption that the quizmaster ALWAYS opens one of the other doors. If he doesn’t, (i.e., if he does not “have to make the choice”) then the whole “it’s better to change your mind” edifice comes crashing down.
Now, here’s another classic probability/game theory question. If the flaw in it has previously been pointed out, I apologize, but I haven’t seen anyone mention it.
It’s the “truel” question, where you have three duellists of varying skills. Mr Black is a crap shot, and hits his target only a third of the time. Mr Grey is a better shot. He hits his target half the time. Mr White is a crack shot. He always hits his target.
To make things “fairer”, the truellists are allowed to shoot in reverse order of competency. Now, the question is: What is Mr Black’s best strategy?
The answer given, without any acceptance of the implicit assumptions involved, is that Mr Black should fire into the air. By doing so, the conventional wisdom goes, he effectively turns a truel into a duel. Mr Grey will obviously aim at Mr White. If he hits Mr White, then Mr Black has first shot at Mr Grey and a 1/3+1/6+1/12 etc chance of surviving. If Mr Grey misses Mr White, then Mr White will shoot Mr Grey rather than Mr Black, thus giving Mr Black a 1/3 chance of surviving. If however Mr Black shoots at Mr White and hits him, then Mr Black has only a 0.5*(1/3+1/6+1/12) chance of surviving. If he shoots at Mr Grey and hits him, Mr Black has a 0% chance of surviving.
So, where’s the implicit assumption that wrecks the maths?
Quite simply, we assume that if Mr Black fires into the air, he misses both his opponents and himself. But wait, we’ve been told that Mr Black is a crap shot! And “the air” here is meant to assume the concept of “infinity”. But, no matter how small Mr White and Mr Grey (and Mr Black) are in relation to the vast expanse of the universe, they are still a positive part of it. Therefore, when Mr Black fires “into the air” there is a finite chance that he will hit someone (including himself). Only for Mr White is there a zero chance of Mr White missing both of his opponents (and himself) if he fires into the air.
Now, in the grand scheme of things, this isn’t usualy going to affect the correctness of Mr Black’s decision (because Mr Grey, Mr White and Mr Black make up a very small subset of the infinity of the universe). But suppose all three of them are in a confined space that is the three-dimensional equivalent of a non-frictional French billiards table? Now if Mr Black “fires into the air” the bullet becomes a random killing machine that will eventually ricochet into one of the truellists. If this is the case, far from improving his chances by firing into the air, Mr Black actually increases the chances of his own death.
So, the implicit assumption in the truellists bit of game theory is that Mr Black might be a crap shot, but he’s a good enough shot to be able to miss when he wants to, 100% of the time. In real life that might be true. In mathematics, it’s a nonsense. The question is not whether the chance that Mr Black will fail to miss is a positive number. The question is “how positive a number is it and how positive does it have to be before Mr Black is better off actually aiming at someone rather than into the air?”
__________
Another sunset on the beach
The promenade. I liked the generation of shadows.
I thought that this picture would come out slightly better than it did. Perhaps it needs some rebalancing of the colours.
Woohoo, segways! The police also use these (as well as bicycles) in the pedestrianised/tramway/bicycle areas of the town.
Place Massena by night.
The western end of Rue de France.
And then I went for a walk with the intention of going into the Vielle Ville. However I was sidetracked by a brasserie – a PMU during the day! – that offered a kilo of moules mariniere and chips for E10.90. Topped off with a fromage blanc with raspberry juice on top (a little too sweet for my tastes, perhaps) and a coffee. And, well, a kilo of moules mariniere! Yumbo yumbo.
On top of that, I discovered that my Citibank card, which I assumed would be useless here because there are no Citibank branches in France, actually works at all BNP Paribas branches. No idea what exchange rate I shall be charged, but it’s a neat way to turn dollars into Euros.
Anyhoo, while pondering these unanswerable maths questions, I thought about an interesting “well-known” probability case that I suddenly realized was flawed.
I spotted the flaw because this is what I do all the time when thinking about poker. What intrinsic assumptions do people make when they play the game? By this I mean, when people say that “Position is an advantage”, what assumptions are made (implicitly) for this to be true? Are there in fact no assumptions? Is position always an advantage? (The answer to that last question, btw, is “no”. From that you can work out why position is usually an advantage. Unfortunately I haven’t yet been able to crack the almighty question of “in monetary terms, how much is position an advantage, given other, known, factors?” But I’m getting there).
People make assumptions in probability all the time. One of my favourites is the “Do you change your mind?” problem. I like it because it catches out just those people who know a bit, but not quite enough.
You phrase the question thus:
“You are on a quiz show. There are three doors. Behind one is a goat. Behind another is a car, and behind the third is nothing. You need to beat three other contestants to get to this point. You beat the other contestants. The quizmaster asks you to pick a door. You pick door C. The quizmaster then opens door A to reveal the goat. Assuming that you want to win the car, do you change your mind?”
The mathematical novice says “it doesn’t matter. You had a one-third chance of being right. Nothing has happened to change that.”
The guy who thinks he’s a bit cleverer than that says to the mathematical novice: “Ah, you donk! Of course you change your mind. By showing you the door that contains the goat, the quizmaster had to make a choice. Therefore it’s twice as likely that the other door contains the car than the door you first chose.”
At which point the man with the plan (let’s call him Jeff Duval, for want of a better choice) says: “Donktastic. No, you do not change your mind. Have you never watched this programme before? The quizmaster only opens one of the doors to reveal a goat or a non-win if you have made the correct choice in the first place. If you have made the wrong choice, he just says “Are you sure? Do you want to change your mind?”
The point here is that implicit in the choice of the guy who thinks he is a bit cleverer than he is is the assumption that the quizmaster ALWAYS opens one of the other doors. If he doesn’t, (i.e., if he does not “have to make the choice”) then the whole “it’s better to change your mind” edifice comes crashing down.
Now, here’s another classic probability/game theory question. If the flaw in it has previously been pointed out, I apologize, but I haven’t seen anyone mention it.
It’s the “truel” question, where you have three duellists of varying skills. Mr Black is a crap shot, and hits his target only a third of the time. Mr Grey is a better shot. He hits his target half the time. Mr White is a crack shot. He always hits his target.
To make things “fairer”, the truellists are allowed to shoot in reverse order of competency. Now, the question is: What is Mr Black’s best strategy?
The answer given, without any acceptance of the implicit assumptions involved, is that Mr Black should fire into the air. By doing so, the conventional wisdom goes, he effectively turns a truel into a duel. Mr Grey will obviously aim at Mr White. If he hits Mr White, then Mr Black has first shot at Mr Grey and a 1/3+1/6+1/12 etc chance of surviving. If Mr Grey misses Mr White, then Mr White will shoot Mr Grey rather than Mr Black, thus giving Mr Black a 1/3 chance of surviving. If however Mr Black shoots at Mr White and hits him, then Mr Black has only a 0.5*(1/3+1/6+1/12) chance of surviving. If he shoots at Mr Grey and hits him, Mr Black has a 0% chance of surviving.
So, where’s the implicit assumption that wrecks the maths?
Quite simply, we assume that if Mr Black fires into the air, he misses both his opponents and himself. But wait, we’ve been told that Mr Black is a crap shot! And “the air” here is meant to assume the concept of “infinity”. But, no matter how small Mr White and Mr Grey (and Mr Black) are in relation to the vast expanse of the universe, they are still a positive part of it. Therefore, when Mr Black fires “into the air” there is a finite chance that he will hit someone (including himself). Only for Mr White is there a zero chance of Mr White missing both of his opponents (and himself) if he fires into the air.
Now, in the grand scheme of things, this isn’t usualy going to affect the correctness of Mr Black’s decision (because Mr Grey, Mr White and Mr Black make up a very small subset of the infinity of the universe). But suppose all three of them are in a confined space that is the three-dimensional equivalent of a non-frictional French billiards table? Now if Mr Black “fires into the air” the bullet becomes a random killing machine that will eventually ricochet into one of the truellists. If this is the case, far from improving his chances by firing into the air, Mr Black actually increases the chances of his own death.
So, the implicit assumption in the truellists bit of game theory is that Mr Black might be a crap shot, but he’s a good enough shot to be able to miss when he wants to, 100% of the time. In real life that might be true. In mathematics, it’s a nonsense. The question is not whether the chance that Mr Black will fail to miss is a positive number. The question is “how positive a number is it and how positive does it have to be before Mr Black is better off actually aiming at someone rather than into the air?”
__________
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How do you move Mount Fuji?
Date: 2009-09-28 03:40 pm (UTC)I never did buy the solution to the three-door problem, and I think the reason is in the rubric. You could add the stipulation that the quiz master always opens a door, and it makes no difference. Whichever door you pick, the quiz master can always pick either empty or the goat.
In the rubric, of course, he picks the goat. This is meant to make you believe that he cannot pick the empty box, because you have already picked it; therefore you swap, and win the car.
In reality, of course, this only matters if the quiz master knows that you are a sexual deviant and are almost as interested in owning a goat as in owning a car.
The answer to my question, btw, and I'm sure I've mentioned this before, is "read it a really sad haiku."