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Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Do any of you have any noteworthy experiences where knowledge of math helped you in an unusual way?
https://en.wikipedia.org/wiki/Monty_Hall_problem
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Get Those Brain Cells Working: The Monty Hall Problem
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(Score: 2) by Flyingmoose on Friday July 29 2016, @10:33AM
In the actual game, the host opens one of the wrong doors to create drama, but the contestant is never allowed to switch doors. Monte Hall himself confirmed that the problem is inaccurate.
(Score: 1) by Demena on Friday July 29 2016, @10:39AM
Just change the name to "The Pick A Box" problem (Australian TV show) and it becomes valid again
(Score: 2, Informative) by Anonymous Coward on Friday July 29 2016, @01:12PM
In the actual game, the contestant is sometimes allowed to switch doors. Monte Hall confirmed that the show manipulated this rule because they knew it increased the odds. (Sometimes they wanted to give away the car.)
For the purposes of the math exercise, the problem must be stated that you are always allowed to switch if you choose.
If you recreate the game with physical objects, the person who plays the host understands the issue within a couple of rounds, even if the contestant person doesn't see it. I find the best way to explain the math exercise here is, instead of showing one empty door and then offering the switch, the host can open no doors, but offer to switch to both remaining doors from the single chosen door. This isn't a change of rules in effect, but it makes the logic clear. The contestant can understand that one of the two doors other doors must be empty, but having two doors is still better than just one.
(Score: 3, Informative) by tangomargarine on Friday July 29 2016, @02:25PM
If he doesn't offer the choice to switch every time, on the "you can have both other doors" offer I would be extremely suspicious I had already selected the winning one. Assuming they wanted to save money and *not* give away the car.
"Is that really true?" "I just spent the last hour telling you to think for yourself! Didn't you hear anything I said?"
(Score: 3, Informative) by WalksOnDirt on Friday July 29 2016, @04:41PM
Sure, that makes it a simple exercise in game theory. You should never switch. The problem has to be stated very carefully to avoid that, and it usually isn't.
(Score: 1, Insightful) by Anonymous Coward on Friday July 29 2016, @06:56PM
That makes sense given your assumptions, but as noted before, sometimes they wanted to give away the car. The sponsor gets better publicity when the contestant wins the car. They have to strike a balance between giving away the car and ending up with a goat to keep the audience interested.
(Score: 3, Interesting) by sjames on Friday July 29 2016, @07:19PM
But the assumption is wrong. They want to give away the car often enough to keep the excitement high. Besides that, the car was given to them by a sponsor and it only holds value for the game as long as it is the current year model. They make a lot more producing the show than they will selling the car off.
(Score: 2) by engblom on Friday July 29 2016, @10:36AM
This is a classic problem and sadly most people fall into the trap, keeping their first door.
Solution:
First when you pick, you have only 1/3 chance to get the car.
When offered to swap doors, you change your odds from 1/3 to 1/2 if you swap the door as totally there are one car and one goat left and there is 50% chance of car in the other door.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @10:43AM
Are you sure it isn't 2/3 change to get the car? You get the combined chance that it was behind the other two doors if you switch.
(Score: 2) by engblom on Friday July 29 2016, @10:44AM
You are actually right! 2/3 it is. Regardless, it is a win situation to switch door.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @02:33PM
Not always. Therein lies the violation of our intuition. If the contestant could intuitively pick the right door on the first try (something our intuition tells us we can do, regardless of the actual facts that the producers have set things up deliberately to make this effectively impossible), then switching would in fact be a lose situation. Statistically this will occur 1/3 of the time.
So, if you're Clint Eastwood in The Good, the Bad, and the Ugly, you may wish to ignore the mathematical aspect of things and go with your instinct.
As Han Solo said, "Never tell me the odds!"
(Score: 4, Informative) by morpheus on Friday July 29 2016, @10:45AM
When offered to swap doors, you change your odds from 1/3 to 1/2 if you swap the door as totally there are one car and one goat left and there is 50% chance of car in the other door.
I may have misunderstood what you are trying to say here but the odds of winning the car if you always switch are 2/3, not 1/2. Just think of trying to pick the door without the car first. Your chances of doing this are 2/3 and if you are successful (and always switch your choice) you get the only door with the car in the end. If you randomly decide between staying with your original choice and switching, then the probability of picking a car is 1/2.
(Score: 2) by tonyPick on Friday July 29 2016, @11:24AM
Yeah - the trick is that since "Monty" always knows which doors have what (and he's always got an empty/goat door) then he's not changing the selection odds in any way from the initial choice when he does the first part of the reveal, even though the question implies otherwise.
Basically the first choice splits the set into your door ("1/3" chance of having the car) and Monty's two doors ("2/3" chance of having the car), and then you get the chance to swap over to Monty's set: when phrased like that the problem becomes a lot more obvious.
(Score: 2) by Reziac on Saturday July 30 2016, @04:26AM
Quick, James -- move the car!
And there is no Alkibiades to come back and save us from ourselves.
(Score: 2) by tangomargarine on Friday July 29 2016, @02:12PM
I still don't get this.
The host must always open a door that was not picked by the contestant (Mueser and Granberg 1999).
The host must always open a door to reveal a goat and never the car.
The host must always offer the chance to switch between the originally chosen door and the remaining closed door.
If the host has to open a different door, the chance is 1/2 regardless of whether you switch between the remaining two doors.
I also don't get the part on the Wikipedia page that separates the doors into 1 group of 1 and 1 group of 2. Consider all 3 doors separately. The only difference is you go from 1/3 to 1/2 with elimination of the third door; you might as well not pick the initial door at all and randomly select 1 of the 2 remaining for all the good it does you.
In an attempt to clarify her answer she proposed a shell game (Gardner 1982) to illustrate: "You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger." She also proposed a similar simulation with three playing cards.
Vos Savant commented that though some confusion was caused by some readers not realizing that they were supposed to assume that the host must always reveal a goat, almost all of her numerous correspondents had correctly understood the problem assumptions, and were still initially convinced that vos Savant's answer ("switch") was wrong.
For somebody the article claims supports the conclusion, it sounds to me like Vos Savant is agreeing with *me.*
So I guess I'm saying both viewpoints are wrong. It doesn't matter whether you decide to switch or not; your odds are the same.
"Is that really true?" "I just spent the last hour telling you to think for yourself! Didn't you hear anything I said?"
(Score: 2) by tangomargarine on Friday July 29 2016, @02:14PM
This whole thing smells like one of those word games where they purposely phrase the rules badly and then laugh at you when you're mislead into picking the wrong answer.
How *wouldn't* the initial placement of the car be random?
"Is that really true?" "I just spent the last hour telling you to think for yourself! Didn't you hear anything I said?"
(Score: 2, Informative) by TheRaven on Friday July 29 2016, @02:28PM
If the host has to open a different door, the chance is 1/2 regardless of whether you switch between the remaining two doors.
You can work it through for each case. Let's call the doors A, B, and C, and the car is behind door C.
If you pick A, then the host must close door B (they have no choice), because it is the only door that did not have a car behind it. If you switch, you win, if you don't switch then you lose.
If you pick B, then the host must close door A (no choice again). If you switch, you win, if you don't then you lose.
The final case, you pick C, now the host can choose A or B, but it doesn't matter, because if you switch then you lose and if you stay then you lose.
The only case where you win if you don't switch is if you picked the correct door first. It works because of the asymmetry. You always gain information when the host opens the door.
sudo mod me up
(Score: 2) by tangomargarine on Friday July 29 2016, @02:45PM
But picking a door to begin with is just showmanship; it doesn't affect the odds at all. If you didn't pick a door at all at the beginning and Monty randomly chooses one of the two goat doors, then you choose a door, you'd still have the same odds.
You always gain information when the host opens the door.
The odds get reduced from 1/3 to 1/2. You don't gain any information about the two remaining doors themselves.
The final case, you pick C, now the host can choose A or B, but it doesn't matter, because if you switch then you lose and if you stay then you lose.
If you stay on the car door you still lose? What?
I'm assuming you mean "choose" instead of "close." Rather confusing typo.
"Is that really true?" "I just spent the last hour telling you to think for yourself! Didn't you hear anything I said?"
(Score: 2) by Gaaark on Friday July 29 2016, @04:46PM
I look at it as:
x. ?
y. ?
z. ?
A door is revealed:
You now have 2 doors with no clue/no information.
Are you REALLY any closer to knowing if you should switch?
I see it as going from 33.33333% repeating chance of having chosen correctly to 50% chance of having chosen correctly. Would switching really increase those odds? Isn't it really just 50%?
Has any information been released to increase your odds?
Not from where i'm standing (looking sharp in my banana suit with my stuffed monkey sitting on my shoulder).
--- Please remind me if I haven't been civil to you: I'm channeling MDC. ---Gaaark 2.0 ---
(Score: 2) by VLM on Friday July 29 2016, @05:25PM
Are you REALLY any closer to knowing if you should switch?
Yeah, but your numbers are too small or percentages are too large making it look weird.
Assume it scales. Scale this to a billion doors. You pick a door which almost certainly isn't it. Dude opens a B-2 doors leaving a winner and a loser. The loser is almost certainly the one you picked, the winner is the remaining unopened door.
(Score: 2) by TheRaven on Friday July 29 2016, @05:56PM
sudo mod me up
(Score: 2) by TheRaven on Friday July 29 2016, @05:54PM
But picking a door to begin with is just showmanship; it doesn't affect the odds at all. If you didn't pick a door at all at the beginning and Monty randomly chooses one of the two goat doors, then you choose a door, you'd still have the same odds.
No, because Monty's choice is different. If you didn't pick a door to begin with then you'd have a 1/2 probability of getting the correct answer, not 2/3. Monty would close one of the doors and you'd be in a trivial situation of having to pick between two doors with no knowledge. By picking one of the doors to begin with, you reduce the number of possible moves for Monty. If you (2/3 probability) pick one of the doors that doesn't have a car behind it, then Monty must close the only other car that doesn't have a car behind it. The other door is then the one that has the car. If you picked correctly the first time (1/3 probability), then the Monty can close either door.
This is a bit easier to explain with a little bit of graph theory and game theory (two of the core bits of computer science, so hopefully not too problematic for people here), but difficult without the ability to draw the relevant pictures. The key point is that if you picked incorrectly to start with then Monty will close the door that doesn't have a car behind it and you then have a 100% probability of getting the car if you switch. For the three doors, A, B, and C, with the car behind C, if you picked A or B then he will close the other, if you pick C then he will close either A or B. The set of doors that are not closed and are not the ones that you picked then becomes:
In contrast, the set of doors that are the one that you picked the first time are:
Picking one from the first set gives you a 2/3 probability of choosing the correct one, picking one from the second set gives you a 2/3 probability of picking the correct one. Or, to put it more simply, if you picked correctly the first time then you win by staying but if you picked incorrectly the first time then you win by switching. You have a 1/3 chance of picking correctly the first time and a 2/3 chance of picking incorrectly, therefore it is better to bet that your first pick was incorrect.
If this is still confusing, I present a simple table of all of the possible paths and outcomes:
From this is should be obvious that there is a 2/3 chance of winning if you switch and a 1/3 chance of winning if you stick.
sudo mod me up
(Score: 2, Insightful) by Anonymous Coward on Friday July 29 2016, @02:31PM
Let's try a different explanation, then.
You pick a door.
One third of the time, you have picked the one with the car.
Two thirds of the time, you have picked one with a goat.
Then the rules force Monty to act based on your choice.
In the 1/3 case, Monty can choose either of the other doors (they both have goats), but it does not matter what he chooses. The remaining door will have a goat.
In the 2/3 case, Monty has no choice - he *must* eliminate the only other door with a goat. The remaining door must have the car.
In the 1/3 case, if you switch, you get a goat
In the 2/3 case, if you switch, you get a car
The reason that it is not 50-50 is Monty lack of choice. No matter what he does, he cannot change the situation you find yourself in when the "do you switch"? question is asked. You controlled that with your first choice ... and there were three options at the beginning.
(Score: 0) by Anonymous Coward on Saturday July 30 2016, @09:49AM
"I still don't get this."
No, you sure don't. I knew there would be at least one of you popping up.
It's simple. There are a billion doors and only one car. You pick a door. Is the car behind it? Of course not. There's a goat behind your door. Now the host offers you the chance to have whatever is behind ALL of the remaining doors. The host even generously offers to take the 999,999,998 goats that are behind those 999,999,999 doors off your hands, leaving you with just the car. So do you accept the offer? OF COURSE YOU DO.
What's that? It's only three doors? Fine - do you want your door and its 1/3 chance or do you want the other two doors with their 2/3 chance of revealing a car? YOU DO IF YOU WANT TO IMPROVE YOUR ODDS.
...Fry
What's that? But the host opened one of those other two doors and revealed a goat? SO WHAT? There will always be at least one goat when two doors are involved. THE SHOWING OF A GOAT TELLS YOU NOTHING. The host is offering you the contents of BOTH doors and he's taking the unwanted goat off your hands, leaving you with a 2/3 shot at the car.
There are NEVER any 50-50 odds. Your odds are ALWAYS the odds of the one door you picked or THE COMBINED ODDS OF ALL THE OTHER DOORS.
(Score: 2) by JoeMerchant on Friday July 29 2016, @02:55PM
And a 50% chance of car in the door you picked the first time.
Україна досі не є частиною Росії Слава Україні🌻 https://news.stanford.edu/2023/02/17/will-russia-ukraine-war-end
(Score: 1) by Mike on Friday July 29 2016, @04:05PM
Actually I think the solution is slightly different.
First, what I'll call the "real politic" solution. If the common wisdom is to switch to the other unopened door, then your choice should be based on whether you think they want to give a car away or not (i.e. have they given a car away lately, how frequently do they seem to give a car away etc...). Basically, try to figure how they are gaming the system.
But assuming total random chance, I think the stats are more along the lines of figure what are the odds that the other two doors are both empty: 2/3 * 2/3 = 4/9. The chance that one of the doors has a car is 5/9. Your choice then becomes 4/9 for the first choice and 5/9 for the unopened choice. Close to 50/50, but it's still slightly better odds to go for the other door.
If you've done some research on the game, though, what I called the 'real politic" solution would probably give you better odds than that.
(Score: 1) by toddestan on Friday July 29 2016, @10:50PM
The "real politic" solution is only in play in the host has the option of revealing the goat and allowing you to switch. In that case, your odds of winning when switching can be as low as 0% (the host only offers to switch when he knows you've picked the car) to 100% (the host is out to help you and only offers you the switch if you've picked one of the goats). But in the problem as presented, the host's hands are tied. He must always offer the option to switch and must always open a door that contains a goat. In that case, the odds of winning the car by switching is 2/3. Which is simple if you think about it, as the chances of initially picking the door with the car is 1/3, and that doesn't change when the host reveals a goat behind one of the other doors, which means that if you stay you have the same 1/3 odds, and therefore switching gives you 2/3 odds*.
"Real politic" might come in handy if you've figured out the placement of the car is not random, or there's a pattern to which goat the host reveals when he has a choice. But that information will only help you so you should still have at least 2/3 odds of winning the car.
(Score: 5, Funny) by Anonymous Coward on Friday July 29 2016, @10:41AM
Because I can't fuck a car.
(Score: 2) by zocalo on Friday July 29 2016, @10:52AM
UNIX? They're not even circumcised! Savages!
(Score: 0) by Anonymous Coward on Saturday July 30 2016, @11:46AM
What are you doing here? Aren't you supposed to be dealing with the aftermath of the failed coup attempt?
(Score: 0) by Anonymous Coward on Sunday July 31 2016, @06:54AM
Don't fucking car e.
(Score: 3, Informative) by engblom on Friday July 29 2016, @10:55AM
import System.Random
attempts = 100000
play :: Int -> Int -> Int -> IO (Int, Int)
play 0 stayWin swapWin = return (stayWin, swapWin)
play attempt stayWin swapWin = do
car <- randomRIO (0, 2) :: IO Int
choice <- randomRIO (0, 2)
if car == choice
then play (attempt-1) (stayWin+1) swapWin
else play (attempt-1) stayWin (swapWin+1)
main :: IO()
main = do
(stayWin, swapWin) <- play attempts 0 0
putStrLn ("Stay winning percentage: " ++ show ( (fromIntegral stayWin)/(fromIntegral attempts)*100) ++ "%")
putStrLn ("Swap winning percentage: " ++ show ( (fromIntegral swapWin)/(fromIntegral attempts)*100) ++ "%")
(Score: 0) by Anonymous Coward on Friday July 29 2016, @12:40PM
I always wanted to learn Haskell.
Now, however, I just understand why a Haskell user would think "Learn You a Haskell for Great Good" is somehow an appropriate title, or sentence in English.
(Score: 4, Interesting) by jdavidb on Friday July 29 2016, @12:59PM
ⓋⒶ☮✝🕊 Secession is the right of all sentient beings
(Score: 3, Interesting) by gringer on Friday July 29 2016, @01:22PM
non-recursive R version, for those who had difficulty understanding the Haskell version:
#!/usr/bin/Rscript
system.time({
attempts <- 10000000;
car <- sample(1:3, attempts, replace=TRUE);
choice <- sample(1:3, attempts, replace=TRUE);
stayWin <- sum(car==choice);
swapWin <- sum(car!=choice);
cat(sprintf("Stay winning percentage: %0.2f%%\n",stayWin/attempts * 100));
cat(sprintf("Swap winning percentage: %0.2f%%\n",swapWin/attempts * 100));
});
Output:
Stay winning percentage: 33.34%
Swap winning percentage: 66.66%
User System verstrichen
0.560 0.092 0.653
Ask me about Sequencing DNA in front of Linus Torvalds [youtube.com]
(Score: 2) by JoeMerchant on Friday July 29 2016, @03:01PM
This statement makes it clear for me:
swapWin - sum(car!=choice);
If you pick the car the first time, then swapping will cause you to lose, but if you pick a goat the first time, then swapping will cause you to win.
Україна досі не є частиною Росії Слава Україні🌻 https://news.stanford.edu/2023/02/17/will-russia-ukraine-war-end
(Score: 4, Interesting) by RamiK on Friday July 29 2016, @11:05AM
This is the first time I saw the wikipedia entry and I have to say, seeing it took a computer simulation to convince Erdős makes me feel a little better about my lack of intuition.
compiling...
(Score: 0) by Anonymous Coward on Monday August 01 2016, @10:03PM
Why do you assume he was good with probabilities?
(Score: 5, Insightful) by ledow on Friday July 29 2016, @11:05AM
I'm a mathematician, by education.
I just work in IT by trade.
I use the Monty Hall problem as a classic example of statistics, probability and human interpretation - why numbers "lie" to you all the time and so you can't just rely on the numbers alone without having the expertise to interpret them.
Monty Hall was the trigger behind thousands of mathematical academics telling the woman with the world's highest IQ that she was wrong. And she wasn't. It's an age-old problem, precisely because it's so simple and yet so unintuitive.
Even as a mathematician, the first time you see it you will be thinking "That can't be right" in your head until you do the maths to convince yourselves.
But it's a great demonstration of why that number in the news - even if completely accurate - shouldn't be the basis of your decision alone. Why you can't just guess at the best course based on instinct. And why you don't mess with statistics or probability unless you truly understand them (gamblers, please note).
The other ones that are great demos of why maths is self-consistent even when it appears not to be, why you don't mess with mathematicians at their own game, and why you shouldn't believe your eyes, is the "rearrange these pieces of a right-angled triangle, oh look we've made a bigger triangle" one. And then anything involving division by zero (those "look, we broke maths" equations that appear to work but contain a hidden division by zero).
Honestly, if people learned just a bit more statistics and probability, gambling revenues would plummet and shampoo ads would end up having to actually confess they are all really just Sodium Laureth Sulphate in varying different concentrations.
Remember, kids, 8 out of 10 cats prefer* it.
(Score: 2) by art guerrilla on Friday July 29 2016, @01:12PM
not buying it: if you euros are saying it is 'maths', when we 'murikans know it as mathematics, and THEN are saying you are 'mathematicians', not gonna allow it...
if you are saying 'maths', should be 'mathers'...
(Score: 2) by sjames on Friday July 29 2016, @06:54PM
Unless they are born in Germany, in which case they are called The Beaver.
(Score: 3, Informative) by JeanCroix on Friday July 29 2016, @01:15PM
(Score: 0) by Anonymous Coward on Friday July 29 2016, @03:54PM
I'm a mathematician, by education. ... Honestly, if people learned just a bit more statistics and probability, gambling revenues would plummet and shampoo ads would end up having to actually confess they are all really just Sodium Laureth Sulphate in varying different concentrations.
No they wouldn't. You are looking at this with a theoretical rational perspective, rather than a practical one. You are taking an psychological/marketing problem and trying to apply mathematics to it.
For the gambling one, there are numerous people who know the math and do it anyway. It's spending money to purchase enjoyment. Think of it this way. "I can spend $20 to see a movie for 2 hours of enjoyment, or I can spend $5 on some lottery tickets and fantasize for an hour over what I would do with millions of dollars." (The strictly rational person would eschew both of them as being irrational wastes of resources.) I suppose you only said revenue would plummet, though, so I guess that could be true as some people are "just that ignorant."
For the shampoo, people are buying an image and lifestyle fantasy. It's the same reason why the Mona Lisa is priceless but an copy is nearly worthless. In all practical ways they are identical, except they aren't. Likewise the Sodium Laureth Sulphate is all mostly identical (excepting that they have different levels of quality control, different amounts of add-ins to change how hair acts, etc), but convey a different feeling upon purchase and use.
Sure it's not rational. However, I'm reminded of the old quote that I heard in an economics class, "only two people act economically rationally: economists and sociopaths."
(I will add that your larger point that "even true numbers told in earnest can be deceiving" is very valid and insightful.")
(Score: 3, Interesting) by sjames on Friday July 29 2016, @07:10PM
On the other hand, I can take an hour walk and fantasize what I would do if I find a winning lottery ticket in the gutter.
I see your point in general, and have to wonder at economists that still cling to the idea of rational actors in a market. It makes the physicist's spherical cow in a vacuum seem perfectly reasonable in comparison.
What would really kill marketing as a profession would be an outbreak of high functioning autism.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @08:57PM
> thousands of mathematical academics telling the woman with the world's highest IQ that she was wrong. And she wasn't.
Some of those probably got the problem second-hand. I know the formulation I got at first was not her full description and had a different conclusion (they hadn't specified he wouldn't open the door I had initially picked IIRC, and said he'd "randomly choose a door with a goat" which retains 50/50 if your door isn't opened and your choice between remainders is 50/50 if your door is opened - yes, really, it's a very different problem). Then their explanation of why it was so didn't make sense and I was left thinking this problem was a stupid fake for yeaaaars until I read the correct formulation, which is transparent to graph theory students. Which, thanks to Martin Gardner, I had unwittingly been since a pre-teen. And I was left SUPER frustrated at the person who presented it to me in the end, as the disinformation was mental muck that took some years to clear.
tl;dr - the monte hall problem is misdescribed sometimes too. :(
(Score: 1, Interesting) by Anonymous Coward on Friday July 29 2016, @02:02PM
As someone who owns over 2 dozen goats, I think I'd rather switch to the door you just showed me gets me another goat. We could always use some new genes in the herd.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @03:46PM
Amen. I'd pick the goat as well.
Especially because if they have the goat on the show, I bet it's a friendly goat. I might even try to feed it a bouquet and give it a nice scratch.
(Score: 5, Interesting) by donkeyhotay on Friday July 29 2016, @03:20PM
I can think of a time when I helped another poor idiot with my knowledge of math. Last year I was in line at a big-box retailer. I bought about $60 worth of merchandise. The clerk rang up the total and I handed her a $100 bill. When she entered the amount tendered, she added a zero, and the cash register told her to give me approximately $940 in change... and she started to give it to me! I said, "No, that's wrong. You keyed in the wrong amount," and she had no clue what was going on. She still wanted to give me the $940! We had to call a manager over to get it straightened out. In the end, judging by the look on her face, she still seemed put out that I had pointed out the mistake. She seemed to have no concept of how much trouble she would have been in if she had come up $900 short on her register.
Come to think of it, there was a time when I did use math to try to benefit myself. A company where I was doing contract programming had one of those contests where they filled a gallon jar full of jelly beans, and then you were supposed to guess how many jelly beans were in the jar. There was some sort of vacation package for the winner. I decided to calculate the approximate volume of a jelly bean and then calculate about how many you could fit in the volume of a gallon jar. When a couple of the employees saw that I was using math, they got pissed. I mean SEROUSLY pissed. I decided it might be best if I excluded myself from the prize, but even when I insisted that I had no intention of taking the prize, that I was just doing it for fun, they were still upset. Eventually, the office manager came to me and asked me to take back my entry card saying, "I know you're just doing it for fun, but it's upsetting people." I just smiled, shook my head, took back the card, and stuck it in a desk drawer. The next day, after the winner was announced, I pulled out my card and showed it to the office manager. I was within 44 jelly beans of the correct answer (which was even way better than I expected). No one had guessed anywhere near that close.
People love math if it is invisible; if it helps build bridges and new houses, helps their cars brake better, keeps them from getting lost -- they just don't want to actually see it. Because when you pull it out and show it to them, even if it is simple, even if you are trying to help them, they think it is witchcraft.
(Score: 1, Interesting) by Anonymous Coward on Friday July 29 2016, @04:12PM
I don't think they were pissed because of some vague hatred/distrust of math, they felt lime you were cheating. Its like setting up a tripod Mount for one if those shooting gallery games. You were ruining the mystery and fun of guessing. I agree they were silly and stupid, but I don't think it was a "witchcraft" kind of thing.
(Score: 2) by DannyB on Friday July 29 2016, @05:56PM
How dare you, a mere fallible human, distrust the perfect machine that says . . . Change Due: $940.00.
That's what she might be upset about.
The anti vax hysteria didn't stop, it just died down.
(Score: 1, Insightful) by Anonymous Coward on Friday July 29 2016, @03:50PM
The phrasing of this problem is itself the problem.
By preloading the terms of the problem with an inaccurate model that does not reflect what is going on, it's not surprising that you're forcing an error on the part of your audience.
Or, to put it another way, it's like saying you're a great hacker because you can do some social engineering. Misleading, at best.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @04:02PM
Nobody wants the poor goat? Maybe it just wants to be loved you insensitive jerks!!
(Score: 2) by DannyB on Friday July 29 2016, @06:00PM
There are people who love goats.
Just like Microsoft loves Linux.
Or Foxes love Chickens.
Or Sharks love Fish.
The anti vax hysteria didn't stop, it just died down.
(Score: 4, Insightful) by MrGuy on Friday July 29 2016, @05:35PM
Here's one way to understand WHY the 2/3 answer is correct.
Take a slightly simplified version of the game, with the following rules. There are three doors. One always contains a car, two are always losers. The player is allowed to pick one door, and the host has the other two doors. At the start of the game, neither the player nor the host know where the prize is. Immediately before the prize is revealed, the host MUST offer the player the option to trade the player's current door for BOTH of the other doors (i.e. the host has no discretion on whether or not to offer the trade).
First iteration of the game - the player picks the door, and the host does nothing else. No doors are opened, nothing is said, nothing. The player picks a door, and is immediately offered the chance to trade with the host for the host's two doors.
I'd hope in this instance we'd all agree the odds from staying are 1 in 3, and from switching are 2 in 3 - you can trade one door for two.
Second iteration. The player picks their door.
Before offering the switch, the host says to the player "Neither of us know where the prize is. However, given we both know there's only one prize, that means at least one of my two doors must be a loser. So, if you trade with me, you're guaranteeing yourself that you're going to get at least one losing door as part of the bargain."
Does this change the odds of winning from the first iteration? It shouldn't. It's still 1 door vs. 2 doors. The host's statement doesn't give you any information you didn't have in iteration 1. Sure, having two doors means having at least 1 losing door. But having all three doors means having TWO losing doors. The game isn't about losing doors - it's having the best chance to have the WINNING door.
Odds of staying - 1/3. Odds of switching - 2/3.
Third Iteration. Player picks a door. Host gives the same speech as before. Then, before the player is allowed to switch, the host goes backstage and sees where the prize is. The host says nothing additional to the player after doing this. Then the switch is offered, as required by the rules.
Does THIS change the odds? It shouldn't - the host is required to offer the switch, so it doesn't really matter if he knows where the prize is or not. The host hasn't given you any new information.
Odds of staying - 1/3. Odds of switching - 2/3.
Fourth Iteration. Player picks a door. Host makes the same comment that he must have at least one losing door. Then the host goes backstage. This time, when he returns, he says to the player "I was right - I just checked, and I definitely have one door that's a losing door."
Does this change anything? Again, it shouldn't - the host just confirmed what you ALREADY know - that one of his doors is a losing door. We've known this since iteration 2. The fact that the host has seen the prize doesn't change anything.
Odds of staying - 1/3. Odds of switching - 2/3.
Fifth Iteration. This is where the trick happens. After the host returns from backstage, the host tells the player "I told you I have at least one losing door." He points to one of his two doors and says "That door is a losing door."
This is where people get hung up, because this is the first time the host has conveyed any information we didn't know in iteration 1. We know know one SPECIFIC door is a losing door.
The trick, however, is that this is actually NO DIFFERENT from Iteration 4. In Iteration 4, the host found out which of his doors were losing doors. Since we've always known the host had one losing door, we've always known he'd have at least one losing door. The host has at least 1, possibly 2, losing doors. Does the host telling us WHICH door is a loser give us MATERIAL information?
Again, it really shouldn't - he's confirming what we already knew. There's at least one door the host has which is a loser.
The host MIGHT have 1 losing door and the prize, or might have 2 losing doors, but in both cases, once the host knows where the prize is, he can definitely point to a losing door.
Odds remain the same - stay, 1/3. Switch, 2/3.
Iteration six. After the host tells you which door is a losing door, he says "Look, I'll prove it to you!" and opens the door he knows is a losing door, revealing there is no prize.
Does this change things from Iteration 5?
Again, it shouldn't. The host already told us that door was a loser. Why does it matter that he "proved" it by opening it?
Odds remain stay 1/3, switch 2/3.
The odds never change, even though we've opened a door. How can that be? There are two keys to this remaining true, both of which apply to the actual game.
First, the offer to switch is mandated by rule. It's not a discretionary action by the host. If the host looked at the doors, saw where the prize was, and then had the OPTION to offer a switch IF he wanted to, then the game is different. In theory, the host should only offer the option to switch if you have the prize - why offer to trade if you can only lose? We're into gambling territory here for what his strategies might be, and whether he's trying to trick you. Lots of options. But the upshot is - if the host has a free choice in whether to offer the switch, then the fact that the offer is made conveys information.
Second, the host does NOT open one of his doors randomly. The host NEVER opens a door with a prize behind it. He always knows, and he KNOWS the door he's opening is a loser. The game would be very different if the host's door was opened at random (i.e. if the host doesn't know what's behind the door he's opening). In that case, there's a 1/3 chance the prize would be revealed by opening the door (every door has a 1/3 odds for the prize before we open them), which makes the solution obvious - you'd ALWAYS trade if you could see the prize behind the host's door. If the host does NOT reveal the prize by opening his door, THEN the odds do indeed become 50/50 we gained new information when a randomly opened door was found not to contain the prize.
(Score: 2) by captain normal on Friday July 29 2016, @08:35PM
Nice WOT, but it totally misstates the actual problem. Please go back and read the Wikipedia link again.
"It is easier to fool someone than it is to convince them that they have been fooled" Mark Twain
(Score: 3, Insightful) by meustrus on Friday July 29 2016, @06:09PM
I always have trouble parsing the solutions to these logic puzzles because the steps are presented out of order to how I think. So here's an attempt to make it simpler.
When the host shows you a door that doesn't win, he is telling you that either your door is right or the door he didn't pick is right. This is important, because which one he is telling you depends on whether you were right to begin with.
1/3 of the time, you were right the first time, so the remaining door is a goat.
2/3 of the time, you were wrong the first time, so the remaining door is a car.
If there isn't at least one reference or primary source, it's not +1 Informative. Maybe the underused +1 Interesting?
(Score: 0) by Anonymous Coward on Saturday July 30 2016, @07:46AM
I'll give you a way to explain it that any middle school kid can understand:
The Packers and the Seahawks are playing on Sunday. Vegas odds tell you, the way the teams are currently set up, the teams are even. Let's say you're thinking of a bet on the Seahawks.
Suddenly, you hear that the star wide receiver of the Seahawks broke a leg. With this new information, you determine that the Packers are the smart pick.
There you are: new information leads to new assessments. Moving on...
(Score: 2) by Open4D on Friday July 29 2016, @09:34PM
Based on the problem as stated in the submission (above), I would stick with my original choice. I assume Monty is employed by the people who have to pay for the car, and I assume that he is only giving me the choice of switching because he knows I have successfully chosen the car. His opening of the other door is just a tactic to distract me from these thoughts.
So I think it's quite important to include the following additional information when stating the problem (described in the Wikipedia article as "standard assumptions [wikipedia.org]"):
Given those assumptions, I certainly agree with the 2/3 answer, and so would accept Monty's offer to switch.
P.S. My favourite line from that wikipedia article is "Pigeons repeatedly exposed to the problem show that they rapidly learn always to switch, unlike humans"! (Though IMHO it should say "almost always". The source given for it is https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3086893/ [nih.gov] )
(Score: 0) by Anonymous Coward on Sunday July 31 2016, @07:18AM
The pigeons are making a decision to switch or stay.
Why whould you say the decision is "almost always" to switch? What evidence do you have for the "almost"?
Even if a few pigeons don't learn to switch "always", then the statement is still true that pigeons (in general) learn to switch, so long as the majority do learn, at least in my book.
Splitting hairs in response to your hair splitting.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @09:35PM
(Score: 0) by Anonymous Coward on Friday July 29 2016, @10:30PM
The original submission was for a slow news day. I guess janrinok felt like posting it anyway. The submission is more popular than I thought it would be.
(Score: 0) by Anonymous Coward on Friday July 29 2016, @10:42PM
Or the answer might be, "people don't know how to read."
The question was, "Do any of you have any noteworthy experiences where knowledge of math helped you in an unusual way?"
53 responses so far. ALL OF THEM about the Monty Hall problem - I read them. NONE about using math knowledge in an unusual way.
I therefore presume the answer to the actual question being asked is, "No." (Even though math has helped me in many ways throughout my life. Not in an unusual way, but rather because I am an unusual person who uses math.)
(Score: 2) by Open4D on Sunday July 31 2016, @06:05AM
I take your point, to some extent. I was guilty myself (though I would argue that the title of the submission - "Monty Hall Problem" with no mention of "noteworthy experiences" - and the way the actual "noteworthy experiences" question is sandwiched between the Monty Hall Problem article quote and the Monty Hall Problem article URL, is not conducive to getting casual readers to focus on the "noteworthy experiences" question rather than the Monty Hall Problem).
But also, you're wrong. The 3 (currently) highest rated comments are not earnest attempts to solve/explain/discuss the Monty Hall Problem. (Admittedly 1 of those 3 was a joke about the problem; but surely everything in the submission is a legitimate target for humour without risking any implied complaints about missing the point.)
(Score: 1, Interesting) by Anonymous Coward on Monday August 01 2016, @02:28PM
Well, I should have also said something to put tongue in cheek as well; re-reading my original makes me totally look like the comment police. (And maybe it was, but I didn't have to be a jerk about it.)
You're right that the article is about the Monty Hall problem - the question almost seems like an afterthought. But individual behavior isn't as interesting to me as the group response to the question. You aren't "guilty" as an individual as much as the collective behavior of everyone was, "Hey! Let's focus on the setup instead of the question," instead of focusing on what to me would have been a more authentic nerdhood of talking about how knowledge of math is a beneficial thing generally and can be slippery.
Maybe instead of Betteridge and Godwin, we need a Monty's Law of Narcissism which states, "any circumstance in which the Monty Hall problem is mentioned will immediately direct all attention to the Monty Hall problem." With corollaries that always exist of: The One Who Tries To Prove It Ain't So, The One Who Notes The Problem Isn't Really How The Show Worked, and The One Who Makes Goat Jokes.
Anyway, your comment deserves kudos.