Fog Creek Software
Discussion Board




The Let's Make a Deal Paradox

http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html

This one drove me insane for about a week.  I went through all the classic stages of grief (except for bargaining).  In the end, this is one of the best examples I have found of how the human mind really sucks at probability.  In fact, when I first read about this in the early nineties the story was that initially mathematicians from real universities (e.g. Harvard) wrote letters claiming it couldn't be true.  Then they did the computer simulations...

Not exactly software project management but I thought the other hyper-nerds might be interested

Erik Lickerman
Saturday, June 07, 2003

It's called the principle of limited choice. Every good Bridge player knows it.

Here's a common application: you have the AK10in one hand and six other cards of a suit in total between you and dummy, leaving four cards out. You play the King as the canons demand and the hand after you drops the Queen and the other hand drops a low card. Now you enter dummy again and lead a low card, the hand before you plays low and you have the choice of finessing the ten and hoping the hand after you started with the singleton Queen or playing the Ace in the hope he started with the doubleton Queen, Jack. It would appear at first sight that the odds are equal, but in fact they are two to one in favour of betting on the singleton queen and taking the finesse. The reason is that if he started with the doubleton queen, jack there is a fifty per cent chance that the would have dropped the jack the first time round instead of the queen, whereas with the singleton queen he would have to drop the queen every time.

It does appear counter-intuitive to start with though.

Stephen Jones
Saturday, June 07, 2003

I've recently been struggling with something far more basic:

The odds of flipping a quarter and getting heads ten times in a row is 1 in 1024. But if you flip heads nine times in a row then the odds of getting heads on the next flip is 1 in 2.

Philo

Philo
Saturday, June 07, 2003

Philo, why is this causing you problems? Or are you being sarcastic?

Stephen Jones
Saturday, June 07, 2003

>The odds of flipping a quarter and getting heads ten times in a row is 1 in 1024.

The odds a flipping a quarter ten times in a row and getting HTTTHTHTTT is also 1 in 1024.

Anonymous
Saturday, June 07, 2003

I'm not being sarcastic - if I've flipped a quarter nine times and gotten nine heads in a row, I find it difficult to believe that the next flip can easily be heads. To me it's counterintuitive.

I mean, intellectually I understand why, but it doesn't "feel" right.

The discussion came up a while ago because a friend used to bet roulette on the binary choices - red/black, odd/even, etc. His strategy was to wait until there were four in a row of one, then bet the other, and keep doubling the bet until he won.

He stopped when someone told him the quarter logic, because he realized there was no mathematical reason he couldn't lose big time.

Philo

Philo
Saturday, June 07, 2003

If you get 9 heads in a row, as opposed to a mix of heads and tails, doesn't it mean it's more probable that the coin is biased.

S. Tanna
Saturday, June 07, 2003

In real life of course if you get nine heads in a row you can bet your bottom dollar the tenth will come heads too, since pure randomness rarely happens.

Which brings me back to Bridge. The odds are there and as in poker you get messed up quick if you don't know them well. Yet they found that in many tournaments the actual distributions were much flatter than the norm. The reason was that the cards were not being shuffled properly from the previous night, and that made for a flatter distribution. With a computer generated distribution you got a greater chance of freak hands, and then people complained the cards hadn't been shuffled properly! It was just counter-intuitive to think that good shuffling produced freak hands, and bad shuffling produced flat hands.

Stephen Jones
Saturday, June 07, 2003

Actually, if you flip a coin 200 times the chance of getting say, a run of 5 heads in a row, is pretty high.  A friend of mine, who also posts to this list sent me an article on that a while ago.  This knowledge can be used to detect falsified data sets.

Erik Lickerman
Saturday, June 07, 2003

And seeing we are on mathematical arcana how about the Birthday theorem.

On average how many people do you have to invite to a party so that there is a more than even chance that two people have the same birthday?

It actually is the theory used to determine the reliability of parity chips since you use the same mathematics to work out the chance of two bits being filipped at the same time.

Stephen Jones
Saturday, June 07, 2003

Erik, for testing for randomness I recommend Benford's Law:
http://www.rexswain.com/benford.html

Philo

Philo
Saturday, June 07, 2003

Ask a friend to draw a spread of random dots on a piece of paper.

Write a computer program to do the same thing using any reasonable random number generator

Compare the results.

S. Tanna
Saturday, June 07, 2003

Well, my distortion of the intuitions of probability is all screwed up.  Grandfather was a professor of economics, and began one of his lectures with a demonstration - given a true coin, he would proceed to flip it such that it landed heads 50 times in a row.

Useful skill, that.

Danil
Saturday, June 07, 2003

I believe Erik was referring to Benford's Law.

Christopher Hester
Saturday, June 07, 2003

I used to play for a chess team whose captain used to toss half-crowns in the navy for bets, and of course alway won.

We would put the four most attacking players on the odd numbered boards so they got white, and the three defensive players with black.

We won the toss every match, and won the league. The funny thing was nobody noticed!

Stephen Jones
Saturday, June 07, 2003

Tanna,

I know what you mean - the person will create an even spread of dots, whereas the random process will produce clustering.

So you are saying that a lack of clustering suggests manipulated results. Very interesting.

X. J. Scott
Saturday, June 07, 2003

"Given a true coin, he would proceed to flip it such that it landed heads 50 times in a row."

Yes it is. It's sort of like baseball players knowing what kind of pitch is being thrown by looking at the direction of the spin of the threads on the ball as it's in midair.

In the book "A Random Walk Down Wall Street" he gives the example of a slimy salesguy predicting wether or not a stock will go up or down.

He sends 500 people a letter saying it will go up. He sends 500 a letter saying it will go down. He then sees what actually happens. He then discards the half he sent the wrong prediction to and send 250 a letter saying a stock will go up and the other 250 saying it will go down.

Eventually, a subset of the population will think he's a genius.

Similarly, he says that if you gave 1,000 stock brokers a coin and had them flip it until they got tails, after 10 flips you'll end up with a lot of people out of the running, but a handful of guys who have the golden touch. The thing is, with the stock market, 10 right moves and everyone will give you their money.

www.MarkTAW.com
Saturday, June 07, 2003

MArkTaw- exactly dude!  Stock brokers, like everyone else, tend not to think of themselves as points on a probability distribution.  I have always wanted to check the records of some of the top stock gurus like Warren Buffet.  My guess is that most of his gains would be attributable to a handful of stocks and that a good stats guy would discover that he is a perfectly acceptible outlyer in a distribution of random stock pickers.  That is over a period of 40-50 years given a million Chimps picking stocks, you would expect a  handful of Warren Buffets, assuming they adopt the buy and hold strategy.

Erik Lickerman
Saturday, June 07, 2003

Buffet insists he only buys stocks when he understands the business.

Considering the way pension fund managers and others threw money to the winds during the dot.com boom, that in itself is enough to put you near the top of the league.

Stephen Jones
Saturday, June 07, 2003

Christopher - No, what Erik is referring to and Benford's law are applied the same way, but they have vastly different roots.

The net rule appears to be "data that looks truly random has probably been faked"

The coin toss issue is based on the idea that truly random data produces clustering.

Benford's Law is based on numerical probabilities based on the incremental nature of counting.

I think.

Philo

Philo
Saturday, June 07, 2003

Does anyone know where to get access to a database containing all the closing prices of all the stocks on the NYSE and or NASDAQ for the past 50 years?  I kinda want to try this now.  My model is a buy and hold strategy with randomly selected stocks.  I figure pick five stocks at random on a random day of the year.  Same day the next year assign a 10% chance that I want to dump a stock that year (i.e. assume I think I am a pretty good stock picker).  If I dump a stock, then I immediately buy a new random stock and do the same thing every year.  Dividends will be automatically reinvested in the shares they came from.

The question is, how many chimps will it take before I generate a few Warren Buffets?  You have to figure that we would learn of these random outlyers because of the American publicity aparatus.

Erik Lickerman
Saturday, June 07, 2003

I don't have a site but I read an article a year or so ago that said they had chimps (literally) throw darts to select stocks.  They competed against the best Stock Gurus and won 3 or 4 years in a row.  Hmmm.

B#
Saturday, June 07, 2003

B# - not quite, it was some brokers who put the WSJ stock pages on the wall and threw the darts. I think those stocks are still performing at about market levels.

You can buy previous stock closings, but you can also get it from finance.yahoo.com if you can write a script to extract that information stock by stock day by day. Well, I think yahoo only keeps a weekly or monthly tally.

I know because I had an idea for an experiment involving stocks that I won't go into here... heh heh.

www.MarkTAW.com
Saturday, June 07, 2003

I would have swore it was monkeys!  Oh well, primates.

B#
Saturday, June 07, 2003

Definately primates.

www.MarkTAW.com
Saturday, June 07, 2003

On the original topic, there's another way to make the "Let's Make a Deal Paradox" seem more transparent (at least to me):

Suppose instead of 3 doors, there were a 1000 doors, with one of them having a prize and 999 having chickens. Suppose you select a door then the host opens 998 of the remanining doors (leaving one alone) and asks if you wish to switch.

Told in this way, it seems clear to me that the chance that you picked a chicken door in the beginning is very high, and the chance that the one other door contains the prize is also very high, so it would seem foolish not to switch.

But again, that's just me.

Steve C.
Sunday, June 08, 2003

A better way might be this.  A chooses a door, and is dismissed from the room.  M reveals all the doors; if the prize is in room three, he moves it to room two - otherwise it stays in place.  Now he closes doors one and two (leaving three open) and has A ushered back into the room.

Danil
Sunday, June 08, 2003

Whlie probability says you switch doors after one has been opened, it also says you probably won't need to worry about it, since the option to switch was offered less than half of the time...

Whapow!
Sunday, June 08, 2003

Erik -
I had the same scoffing, "That can't be true!" experience with that puzzle, too.  The variation mentioned by Steve C. was what finally made me repent and submit to the truth.
In fact this was the event that made me appreciate  Maralyn Vos Savant, that woman who writes a column for Parade magazine.  She got this right and endured a lot of  grief from experts who thought she was wrong and an innumerate fool.
Hey, how's Buggs?

Ethan Herdrick
Sunday, June 08, 2003

The high IQ lady was indeed the one who brought this up but my recollection is that her explanation was actually wrong.  I think she may have stumbled into it.  On the other hand I read her explanation in the NYT so it was at least second hand and the author of that article may not have understood it.

Erik Lickerman
Sunday, June 08, 2003

Here's that article in the NYT:
http://216.239.39.100/search?q=cache:Dyd4-ihd3kMJ:www.dartmouth.edu/~chance/course/topics/Monty_Hall.html+&hl=en&ie=UTF-8

The only nitpick they have with Vos Savant (is that a real name?) is that she failed to specify that the game show host must be required to flip a door.  If it's up to him, then you have to figure out if he likes you or not, etc.  But obviously she has to conserve words in a column in Parade.

Ethan Herdrick
Sunday, June 08, 2003

The strange thing, as I said before, is that this has been known by Bridge Players for at least 40 years. Terence Reese, published it in "The Expert Game" which dates back to the 60's I think.

Perhaps we ought to start a JOS cards club to improve our Math

Stephen Jones
Sunday, June 08, 2003

Doesn't windows come with a network bridge game built in now?

www.MarkTAW.com
Sunday, June 08, 2003

There are plenty of places you can play Bridge on the Internet, though I suspect it would be somewhat of a slow process compared to the real thing.

I haven't noticed it on XP; it would be bye-bye to productivity if it's on the machine at work.

In general, unlike chess software, Bridge software is atrociously bad. Whether this is because chess software has had tents of millions of dollars and some of the world's best brains invested in it since the 1950's, or whether it is because the co-operative nature of Bridge is anathema to computer programmers is a moot point.

Stephen Jones
Sunday, June 08, 2003

+1 for the sad state of Bridge software. I was determined to learn, and read a few books (the best of which I thought was Bridge for Dummies, but still, none of them treated it like an engineering problem). Nobody local to play with, so I looked for computer opponents, and the software is terrible. I suppose I could play on Yahoo, but Bridge is such a partner focused game that I'd find it hard to play against people of such widely varying ability.

Oh well. :-p

Brad Wilson (dotnetguy.techieswithcats.com)
Sunday, June 08, 2003

Two thoughts on Windows cards software:
1) Anyone notice they actually fixed solitaire? When you win a hand, the cards bounce off the screen. In Win95, the bouncing got faster as the CPU got faster. When I installed Win2k on a 2GHz machine, I gleefully fired up Solitaire to watch how fast the cards sprayed and...
The bouncing was back to a leisurely "normal" speed, and is obviously set to the clock instead of the CPU now.

I am amazed that someone in the Win2k development effort actually went back into the Solitaire code and took the time to "fix" it.

2) I'm still convinced that Windows Hearts cheats. :-)

Philo

Philo
Sunday, June 08, 2003

As regards stock picking strategies, being the cynic that I am I have long suspected that it matters not a fig whether your pension fund manager is an ape or an analyst.

The true secret of stock market sucess, I believe, is to have a very large reserve of cash and a high tolerance for failure.

Still, it'd be interesting to test my theory on some real market data and a random strategy.

Chris Davies
Monday, June 09, 2003

"How do you make a million in the stock market?"

"Start with two million."

www.MarkTAW.com
Monday, June 09, 2003

OKBridge has been for me generally faster than club play, once a foursome is established.  No dealing, no waiting on other tables, etc.  My experience somewhat biased by having a team I meet on a weekly basis (which normally means three of us show up at a set time, and find a fourth in the crowd).

For a computer opponent, I haven't looked for a while, but I suspect GIB is still the best choice
http://www.gibware.com/

Danil
Monday, June 09, 2003

Ethan

Not to beat a dead horse, but the article makes no mention of Vos Savant's original explanation- only that she was correct and then, at the bottom, an attempt at an explanation.  This is why I suspect my original memory may be correct, that she was right but her explanation was wrong.  I am a bit more cynical.  My guess is she tested the probability experimentally before answering; something her math critics should have done

Erik Lickerman
Tuesday, June 10, 2003

The Parade article in question was September 9th 1990, and Parades archives are only searchable since the beginning of this century.

I have however come across this apparent resume of her original explanation, and it does appear she explained it perfectly, and it is clear that Dr. Sachs was very, very wrong.

"---"Yes, you should switch." She explained that, obviously, there is a one-third chance that the original choice, door No.1, is the correct one and there must be a two-third chance that the car is behind door No.2 or No.3. Since door No.3 has now been eliminated as a possibility, the second door has a two-third chance. To make her point more strongly, she wrote: " Here's a good way to visualize the problem. Suppose there are a million doors, and you pick door No.1. Then the host, who knows what's behind the doors and will always avoid the one with prize, opens them all except door No.777,777. You'd switch to that door pretty fast, wouldn't you? Since you choose first, it's unlikely that you picked the door that hides the car. The same logic applies whether there are three doors or a million doors." ---

Stephen Jones
Tuesday, June 10, 2003

Okay.  That is a good explanation.  I read it second hand in the first place so most likely the reporter who conveyed it didn't quite understand it.

Erik Lickerman
Tuesday, June 10, 2003

Once I finally understood the problem I was shocked at how obvious is was.  I guess my mind has been clouded regarding probability by all training.

Here in the UK there used to be a kids show called beat the teacher.  It involved a pupil of about 10 or 11 going up against there teacher.

Thing is both the child and adult had the same questions to answer, where werent differentiated.  Suprising thing was that he children normally won.

The reason was that the questions were of this counter intuitive type.  The children found them easier because there minds are so much more flexible.

I bet The Real PC could make something of that :)

Ged Byrne
Wednesday, June 11, 2003

*  Recent Topics

*  Fog Creek Home