Game Show Probability: Newsletter Challenge (05/01/07)

Right ON!

> Think of it like this: We have doors X,Y,Z. You pick door X and some guy in the audience picks door Y. The host opens up door Z and reveals a goat. This talk of switching doors to improve your chances of winning would guide you to switch from X to Y and the guy in the audience to switch from Y to X... hmmm... how can door Y be more probable for you but X more probable for the guy in the audience?

=-==

Here in lies the fallacy of the "it don't matter" crowd.

The questions states that the host [who picked "Z" in your example] ALWAYS [or at least, in this case] picks a goat.

So, AFTER you picked X, then the host HAD to choose the goat. 1/3rd of the time, he could have picked Y OR Z. And 1/3rd of the time, he Had to pick Z. And 1/3rd of the time he Had to pick Y [outcome was not the same in this 1/3rd of the cases for you and the guy in the audience.]

=-==

> how can door Y be more probable for you but X more probable for the guy in the audience?

=-==

Simple, X is not more probable for the guy in the audience!

The host was constrained in his choice by the choice You made but not by the choice of the guy in the audience. Or,

If you and the guy in the audience both chose a goat [1/3rd of the cases possible], then the host must choose the goat that the guy in the audience chose; thereby eliminating Him from the game, but Increasing your chances of winning, if you switch [remember, you and the guy in the audience both chose a goat in this case; which IS the case in 1/3rd of the cases possible]

=-==

tom

Nearly all gamblers think they understand statistics better than the next man. If the next man is another gambler, precisely half of them do. But, at best, the reason they are gamblers is that any theoretical understanding of statistics doesn't extend to their actions. If they followed up on a proper understanding of statistics, they would be card-sharpers or equivalent, not gamblers.

There are several different answers to this question depending on the algorithm that the game-show host adopts - the question implies but fails to state what this is. You can give any of the three conceivable answers - (should switch, should not switch, doesn't matter) - provided you state your assumptions.

A categorical answer without stating your assumptions can only mean that you either didn't read the question carefully enough, or you don't really understand statistics

OK, I think I understand the cause of the great disparity in answers and the steadfastness with which most of you appear to uphold your own solution.

The problem, as often is the case in these challenges, is that all constraints are not spelled out and some assumptions must be made to arrive at a discrete answer. Just saying "it depends" does not cut it, though.

This "Challenge" question does NOT say anything about the host's motivation, knowledge or lack thereof of what is behind the doors, or, and especially, whether or not the host will ALWAYS offer you a second chance, however this is sort of implied by the wording. So we ASSUME in computing the solution, or solving by experiment, that in subsequent games the host will always offer you a second chance to choose another door or keep the first on picked.

Maybe the host only offers a second chance because he knows you have picked a winner the first time. In that case, if you switch, your probability of winning just went down from 100% to zero! The point is, you don't know if he ALWAYS will provide a chance to switch, will do so randomly, or will only do so to insure you lose! Who knows, he may allow you to switch because he WANTS you to WIN!

Hmmm, maybe I will submit an ambiguous "Challenge" question. They seem to be the most fun and offer the most diverse interpretations and solution!

Indeed - that is why it's important to state assumptions for this 'challenge'. However, not all the variety seems to be down to that.

This "Challenge" question does NOT say anything about the host's motivation, knowledge or lack thereof of what is behind the doors, or, and especially, whether or not the host will ALWAYS offer you a second chance, however this is sort of implied by the wording. So we ASSUME in computing the solution, or solving by experiment, that in subsequent games the host will always offer you a second chance to choose another door or keep the first on picked.

Interestingly, you seem to believe that you need more information to determine the probabilities. You don't. For example, whether or not the host will "ALWAYS offer you a second chance" has no bearing at all on your current situation. How could it? You have one shot; you are on stage now, not tomorrow. His offering someone else a different deal will not change your chances.

The question "does NOT say anything about the host's motivation" nor does it need to. How can that help? Suppose his motivation in showing you a goat was to confuse you. Suppose it was to help you. Calculate the probabilities both ways, and tell me what they are.

Maybe the host only offers a second chance because he knows you have picked a winner the first time. In that case, if you switch, your probability of winning just went down from 100% to zero!

You can't really mean this can you? Are you suggesting that there is a rule known to the contestant and the host that says: If the contestant picks correctly, then and only then the host must offer the contestant a second chance. Wouldn't that seem a little pointless?

All challenge questions are ambiguous. Language is inherently ambiguous. Actually, the CR4 people did an excellent job editing this question down from a more explicit version, leaving in just the essential details. Their version is much better than my original submission. The other stuff (motivation, whether the host always offers a second chance, etc.) only gives the illusion of disambiguation. In fact, it matters not. To prove that to yourself, calculate the probabilities each way. If you have trouble , ask Fyz for help -- he seems to have a handle on this probability stuff.

"seems to have a handle on this"... I'd be pretty limited doing experiments in modern physics if I didn't.

However, I would support STL Engineer on the fact that the host's knowledge and procedures are important.

I'll just cover three of the clearer cases:

1) Host always opens one of the two unchosen doors, and always reveals a goat: this is the 'classic' case, and the proportion of wins is 2/3 if you change your choice, 1/3 if you don't.

2) Host opens one of the unchosen doors, but has no idea what is behind any of the doos, and happens to reveal a goat: the proportion of wins is 1/2 whether or not you change your mind. (If he knows nothing, it doesn't matter whether he always does this or it is just on this occasion)

3) Host opens a door if and only if you choose the door with the car; proportion of wins is zero if you change your mind, 1 if you don't.

My assumption was 1) above - that this was an ongoing game show and you knew that this always happened. If you don't have such knowledge of the case for the particular show, then you need an idea of the probabilities of the different host behaviours. As STL Engineer implies, you can in principle use any knowledge you may have of the character and motivation of the particular host to assist in this. Without that, you should be looking at the overall probabilities of different host behaviours, and applying conditional probabilities to the results. BTW, recent phone-in scandals in the UK have shown how pressures can modify behaviour.

That's way beyond what is practical here - and having over the years helped review papers in experimental psychology for a member of my family, apparently way beyond the capability of the professionals too.

Regards to you both

Fyz

I have to say I admire your ability to persevere in explaining the situation Fyz . I haven't counted , but it must be a lot . I wonder what your family member (experimental psychologist) would make of these questions . ie analyzing % of people getting it wrong/right , commenting before reading thread , number changing opinion , etc. Some time I will have to print off an entire thread and try to figure some stats out , but that may require a lot of analysis. Kris

> 2) Host opens one of the unchosen doors, but has no idea what is behind any of the doos, and happens to reveal a goat: the proportion of wins is 1/2 whether or not you change your mind. (If he knows nothing, it doesn't matter whether he always does this or it is just on this occasion)

No, it doesn't matter whether he knew or not. The fact is that he did pick a door with a goat, and the odds go up if you switch...

contriwise, if he had "no idea what is behind any of the doos, and happens to reveal a" car, then your odds went down to 0

=-==

ACCORDING TO MATHS:

we have 3 doors so let's name

door with Goat1=G1

door with Goat2=G2

door with Car=C

whatever door the player picks the host shows him a goat now.NOW there two cHances to be G1 or to be G2

so we have the chances of G1 being first

player door - remaing door

G2 - C

C - G2

CHANCE OF CAR => P(C)=N(C)/N(W)=1/2=50% IF HE PICKS THE OTHER DOOR IT'S

P(C)'=1-P(C)=1/2=50%

and the G2 chances

G1 - C

C - G1

CHANCE OF CAR => P(C)=N(C)/N(W)=1/2=50%..PICKING THE OTHER DOOR =>

P'(C)=1-P(C)=50%

SO WHATEVER THE PLAYER WILL DO HE WILL HAVE THE SAME CHANCES TO WIN THE CAR

I think you say that host always shows you a goat, and behind a door you did not pick. Unfortunately, your maths appears to correspond to the situation where host may show you goat or car at random, and happens on this occasion to show you a goat. The maths are not the same. Go away and try it experimentally.

Fyz

FYZ!

you've hit the nail on the head my good man! I can't believe some of the posts here.

Ok HERE IS THE ANSWER!!!...and REMEMBER, The host will ALWAYS open a second door to reveal a goat.... it says so in the original question!! You MUST choose one door at the begining. The main issue here is the choice you have of either sticking OR swapping...NOTHING ELSE!!

OK?

There are three doors. Let's call them A,B and C and, for the sake of argument, let's assume that the car is behind door C.

You pick door A, host MUST open door B. If you stick, you get a goat

You pick door B, host MUST open door A. If you stick, you get a goat

You pick door C, host opens door A OR B. If you stick you get a car

Therefore if you stick you get a car 1/3 of the time.

Same scenario as before, the car is behind door C

You pick door A host MUST open door B. If you swap, you get a car

You pick door B host MUST open door A. If you swap, you get a car

You pick door C host opens door A OR B, If you swap, you get a goat.

Therefore you get a car 2/3 of the time by swapping.

Now, for the slow ones among us, try putting the car behind any of the other doors and go through the same process all over again. Remember, the host will ALWAYS show you a goat! The question states this clearly. It doesn't mention any reason for showing you a goat, merely that he does. What a funny fellow he is!

As I always say to my daughter, if the question seems unclear at first, read it again and again, until it does. . . If it doesn't, then take up art!!

Alun H

Alun H

I think we have to think that the host is honest and just selects a diferent door that is selected by the contestant, if we accept this the solution is in comment Nº 185

Ciro

On the assumption that the host always opens another door, and the one he selects has the goat, you should always change. Unfortunately, the text in #185 is not clear enough to identify where it went wrong (and it may just be the words that are wrong , and not the intention). Please see my post #206 for more complete description of weightings.

Dear Physicist:

Because a drawing values more than a million words, I did this decition making tree, in witch you can see that you have 2/3 posibilities to fall in the yellow branch while the red one has 1/3

So it is more probable to fall in the yellow case, then to get the better chance we have to stay in our original selection when asked for the second trial as you can see en the graph.

CIRO

Hi Ciro:

I admire your industry in creating the chart! I'll let Fyz respond more specifically (in hopes that he might be more familiar with the chart format*), but I wanted to suggest that you might use a simulator to see how the two possible strategies work. Here's a link.

*To me, it looks like you may be thinking that the host, after hearing your second guess, once again opens a different door than you asked for. (Given that there are only two doors left to chose from, the logic gets a little peculiar.) Your second guess (in other words, staying with your first guess or switching) ends the game: whichever door you end up choosing is opened, and you win what is behind it.

Hi, Blink

Yes, I make some assumtions before starting, I did the following assumtions:

1.- The host knows nothing about were the goats and car are place behind the doors

2.- The host acts honestly and always opens a different door than the selected by the contestant

3.- In the firt trial he has to open one of the two posibilities ( one of the two not selected doors)

4.- In the second trial he has only one door to open the one not selected

This are all reasonable assumtions reading the original problem.

As you sujects, I add numbers to the diagram and I made an explanation as follows:

It is important to take care with the words OPEN and SELECTS

1. Is one off the chances in the first trial, the possibilitie to selects a door with a goat

P(G1 or G2) =2/3

2.- If we are in the situation (1) the host could open the door with a goat or with the car (not the case acording to the problem)

He opens a door with the goat

3.- Been in this situation the contestant can stay in the selected door (a goat) or change to the other door (the car), remember he is pointing to door wit a goat and the other door has the car, if he stay and acording to the assumtion the host opens the other door the contestant will win this exactly what i recomend.

4. and 5 are the subsecuent stages following the decision made

In this branch ( the yellow one) we get the car if we "stay" in the selected door, this branch has a possibility of 2/3

6. Is the other chance in the first trial, the possibilitie to selects a door with the car

P(Car) =1/3

7.- If we are in the situation (6) the host could open the door with the goat1 or goat 2

He opens a door with a goat anyway

8.- Been in this situation the contestant can stay in the selected door (the car) or change to the other door (a goat), remember he is pointing to the door wit a car and the other door has a goat, if he stay, acording to the assumtion the host will open the other door, then the contestant will loss, if he change he will win, in this case I should recomend him to change, but the possibility to fall in this branch is only 1/3

9. and 10 are the subsecuent stages following the decision made

In this branch ( the red one) we get the car if we "change" the selected door, this branch has a possibility of 1/3

FINAL CONCLUSION

It is more problable that we fall in the yellow branch ( 2/3 ) so we recomend the contestant to STAY

I hope all this clears my point

Regards

CIRO

Hi Ciro:

Yes, that clears it up. As this problem is usually presented, the contestant (you) are asked to choose one of three doors. The host does not immediately open the door you chose, but instead opens another door, and to avoid giving away the game entirely, he does not open a door with a car behind it. Therefore, he must know where the car is.

By showing you the location of one of the goats, he has given you an advantage (over simply ending the game with your first guess, and giving you whatever was behind that door). He asks if you would like to switch from your first guess, or stay with that first guess. Many contestants would , at this point, think there is no advantage in switching: they view the choice as 50-50. When you make your decision to switch or not, the host opens the door of the original guess (stay) or the remaining door (switch). In other words, the host does not again open a door other than the one chosen.

So, I think you are just operating under different assumptions.

Ken

Hi Ken

Ok, if in the 2nd trial the host opens the chosen door, then the recomendation is to CHANGE the door, this is small change at the end of the play that makes it easy and no need to assume strange performace of the host and avoiding complains when the contestant selects the door with the car and the host opens the other.

It is a better set up of the game

nice play

regards

CIRO

Hi Ken

There is no reason why he can't, as you put it, "give the game away" 1/3 of the time that he opens a different door. He just has to have a suitable line of chat to keep the audience on side when this happens.

In any event, if host knows nothing, the probability after he has opened a door with a goat becomes 50% (i.e. still 1/3 from the original situation for the door you chose), not the 2/3 that the sequences fail to demonstrate..

Fyz

You must make some assumptions before you start. They can be that the host knows nothing and just opens a door, or that he knows and always opens a door other then yours that hides a goat.

For those cases, starting at the left, the first three columns look correct, showing that if your first guess is a goat-door (2/3), then your switch will be to the car-door; and if you guess car-door (1/3), switching will give you a goat-door. So: if you want a car you switch (gives 2/3 for the car). I can't make out what you are trying to say after that - possibly a system incompatibility; but it looks as if you are suggesting that you don't get whatever is behind the latest door that you choose, which is the reverse of what would be expected. If I've got this wrong, please explain what you meant (possibly number the boxes and write descriptions against the numbers in unabridged text?)

Fyz

Phisicist and Fyz

Yes, I make some assumtions before starting, I did the following assumtions:

1.- The host knows nothing about were the goats and car are place behind the doors

2.- The host acts honestly and always opens a different door than the selected by the contestant

3.- In the firt trial he has to open one of the two posibilities ( one of the two not selected doors)

4.- In the second trial he has only one door to open the one not selected

This are all reasonable assumtions reading the original problem.

As you sujects, I add numbers to the diagram and I made an explanation as follows:

It is important to take care with the words OPEN and SELECTS

1. Is one off the chances in the first trial, the possibilitie to selects a door with a goat

P(G1 or G2) =2/3

2.- If we are in the situation (1) the host could open the door with a goat or with the car (not the case acording to the problem)

He opens a door with the goat

3.- Been in this situation the contestant can stay in the selected door (a goat) or change to the other door (the car), remember he is pointing to door wit a goat and the other door has the car, if he stay and acording to the assumtion the host opens the other door the contestant will win this exactly what i recomend.

4. and 5 are the subsecuent stages following the decision made

In this branch ( the yellow one) we get the car if we "stay" in the selected door, this branch has a possibility of 2/3

6. Is the other chance in the first trial, the possibilitie to selects a door with the car

P(Car) =1/3

7.- If we are in the situation (6) the host could open the door with the goat1 or goat 2

He opens a door with a goat anyway

8.- Been in this situation the contestant can stay in the selected door (the car) or change to the other door (a goat), remember he is pointing to the door wit a car and the other door has a goat, if he stay, acording to the assumtion the host will open the other door, then the contestant will loss, if he change he will win, in this case I should recomend him to change, but the possibility to fall in this branch is only 1/3

9. and 10 are the subsecuent stages following the decision made

In this branch ( the red one) we get the car if we "change" the selected door, this branch has a possibility of 1/3

FINAL CONCLUSION

It is more problable that we fall in the yellow branch ( 2/3 ) so we recomend the contestant to STAY

I hope all this clears my point

Regards

CIRO

oh dear... Can I point everyone back to the original question for a moment...

"On a game show, a contestant is offered the choice of three doors. A goat is behind two of them, a new car behind the third. The contestant picks one door, but the host opens a different door showing a goat. With two doors left, the host again asks which door? To maximize the probability to win the car, should the contestant stay with the first guess, or switch to the other door? Explain the reason behind your answer."

Ciro you said...

Yes, I make some assumtions before starting, I did the following assumtions: 1.- The host knows nothing about were the goats and car are place behind the doors 2.- The host acts honestly and always opens a different door than the selected by the contestant

well this contradicts the original question!!

i.e."but the host opens a different door showing a goat."

So, if we assume that the host does what is suggested in the original question (because that's all we have to go on) then your entire diagram is (however beautiful to behold) pointless!

please refer back to my post No.183 for the solution to this problem

Sorry if this post seems blunt but I've been at the wine again!!!

phzzblggrurp!! pardon me!

Alun H

Hi, Alun H

Sorry, but I dont see any contradiction:

1. The original does not say anything about the knowledge of the host

2. Also does not say anything about the host been dishonest

3. The original says "but the host opens a different door" I only add the word ALWAYS which is a valid assumtion and not cortradicts anything

¿Where is the contradiction?

Anyway, perhaps the peruvian Pisco Souer is making me confused

Regards

CIRO

Ciro

I see nothing wrong with your original assumption, as the word "always" is not used in the question; but your conclusions are incorrect. If the host knows nothing, the chances are 50-50 whether you change or not - and it doesn't matter what his behaviour is on other occasions. (In fact, the behaviour on other occasions is only relevant to understanding the algorithm host uses on the specific occasion in question)

If you have selected the car (1/3), then host has a 100% chance of selecting a goat - so that situation accounts for 1/3 of the initial conditions.

If you have selected a goat (2/3), the host has a 50% chance of selecting another goat. So this condition also accounts for 1/3 of the initial conditions.

The other 1/3 are accounted for by you selecting the goat and the host selecting a car - at which point you already know you have lost.

I trust that this is clear.

Fyz

Physicist

The important thing is that it is more probable that the contestant falls in the yellow branch than in the red one and if you fall in this situation you should select the door accordingly, switch if you do not use the word always or stay if not, anyway you will get the car with 2/3 probabilities.

Lets wait for the answer tomorrow

Regards

CIRO

Unfortunately, repeated assertion does not make this true. Let me try to explain the proper result again. If you don't agree, please don't merely restate your present position, but show me where the error is.

When you choose initially, the probability that you have chosen the car is 1/3. If you have the car, the conditional probability that host will open one of the two goats is 100%. That accounts for 1/3 of the possibilities.

The probability that you choose one of the two goats is 2/3. In that case the conditional probability that host will open a door with one of the two goats is 50%. This accounts for a further 1/3 of the cases.
On the other hand, the probability that host will open a door with the car is also 50%. that accounts for the final third of the cases. This third of initial cases is not included in the situation we are describing.

Clearly, as described above, the probability of your initial choice concealing the car is equal to that of it concealing a goat. That means 50% for each case.

It wasn't clear from your text, but I suspect that you were assigning 50% of initial cases to host showing the car, rather than 50% of the conditional cases. That would give the numbers you are proposing.

A final note - overall, having no knowledge, host must opens a goated door 2/3 of the time. If that makes your probability of getting the car by sticking with the original door 2/3, then the overall probability of your having getting the car can be increased from 1/3 to 4/9 by virtue of the host opening a different door and then your door - and without giving you the choice to switch. That sounds like white magic to me.

Regards

Fyz

Hi Physicist

I think that a good a argument is the one given by Blink in the answer, particularly in the paragraph as follows:

"Perhaps the simplest explanation is this: The probability that you picked a goat on your original guess is 2/3. Few will argue with that. So, 2/3 of time, when Monty shows you the location of the second goat, you're home free – the goat must be behind the remaining door. 2/3 of the time, you will win by switching. "

To me it is a good and short explanation of what I was pointing.

Regards

CIRO

Hey - that's familiar Ken ! I will sue you know.

For those wondering why Kris is threatening to sue:

The topic of the Monty Hall Paradox came up in a thread on wacky science (which, BTW, may have set a record for number of posts). In that thread, Kris offered an explanation of the probabilities involved that I liked. Of course, his explanation lacked the literary spark and flow of mine, and, if I recall, was full of grammatical errors, misspellings, errors of fact, gross generalizations, ambiguities, lies, and distortions. But other than that, it was very similar to one of my explanations here. Further, if I recall correctly, his explanation did not contain the error that mine does, in which I use the word "goat", when intending to write "car."

That wasn't the agreed wording ! My solicitors will retain the photographs for future use. Your behaviour so far on the Gyroscope thread is acceptable , but tread carefully. All you others who are in my album be warned by Kens folly . I will not tolerate failure to incite mischief.

Hi Ciro

Ken's posted explanation was great - with the exception of a mistype. He typed "the goat must be behind the remaining door", when he meant* to type "the car must be behind the remaining door". I expect the problem is that it takes a long time to get a correction via the editorial system - if indeed the software allows it at all. In any case, his following sentence says you win 2/3 of the time if you switch, which is the same as I am suggesting (assuming that winning means you get the car).

Answer me this: if you initially selected a door that concealed a goat, and host revealed a goat behind a different door. What must be behind the third door?

Fyz

*BTW, that is not merely my opinion - he admitted he found it hard to believe that he had failed to notice the error when checking the text. I often have a similar problem when proof-reading my own stuff - I read what I meant to write, rather than what is actually printed.

Hi Physicist

The answer to your question is obviously the car, but you are starting with a fact that is not in the problem "if you initially selected a door that concealed a goat" in the real problem you can not say that you only have to talk about probabilities.

Anyway, I am already happy with the solution, let look for other challenges to exersice our brains.

kind regards

CIRO

The "if..." was covering two thirds of all cases; "if" is not given as a fact, but a definition of the options that are currently being considered. This was clear in the context of addressing the mistype in Ken's/Kris's explanation. As the other third were explicit in that explanation, it is incorrect to call this "a fact that was not in the problem".

But hopefully it was a problem with your English, and you were referring to something else?

Fyz

That's hilarious , I only just noticed the typo ! The explanation method was mine to Ken , but in fairness the header states that the suggested question was modified after Ken proposed it (i think , since I can't check in the editor box - which is another cause of slips ) so the error is probably that of CR4 editors. They can modify a post at any time , us mortals have 15 Min's to hit the edit button. I think the 'not - seeing' issue is the same as the oft used 'count the number of times e occurs' poser.

It's funny... I think most people read the explanation as if the typo was not there. I just checked -- the typo is indeed mine, and not caused by the CR4 editors -- in fact, the explanation text was just copied in, I assume. Had they edited it, they probably would have caught the typo.

The original question was edited down considerable by the CR4 people, leaving out the assumptions, but I think that worked fine -- the assumptions are pretty easy to state as part of the answer, and most people seemed to fill in the details as expected.

BLAME THE EDITORS KEN ! SUE THEM !

< I want a cut >

The Question worked well - Interesting stuff always comes out when a question is nit-picked to death. I'm sure CR4 members could give a Piranha a run for it's money !

Alun H

you look very confident when you keep your CAPS lock on.

here's where there's something not quite right in your "ANSWER": same assumption as A, B, C doors with the car behind the C door.

u pick A, host open B, u stick u get goat
u pick B, host open A, u stick u get goat
u pick C, host open A, u stick u get car
u pick C, host open B, u stick u get car

same goes for switching logic. therefore its 4/8 = 1/2 each. thus the odds are even, doesnt really matter.

cheers.

You should have wrote:

u pick A, host open B, u stick u get goat
u pick B, host open A, u stick u get goat
u pick C, host open A or B, u stick u get car

To get to pick C twice, you would have to play the game twice.

how come u'r merging the third instance which is leading to two possible situations. u should count all of the possible cases. the originial statement which was made is:

if u picked A and host opens B (which he should do)
if u picked B and host opens A (which he should do)
if u picked C and host opens A (here its his choice)
if u picked C and host opens B

"If" is the condition on all four. U can see its not about playing twice, or three of four times. u'r just making one choice but u have to count anything possible, which leads with 2 out of 4 positive.

Hi Shad

You are assuming that each of your four options occurs with the same frequency. That is simply wrong. You could have chosen any of three doors in the first instance. The initial options are as follows. I am placing the door you select at the left, and the number at the right is proportion of the tries that each version will occur:

G1 G2 C 1/6
G2 G1 C 1/6
G1 C G2 1/6
G2 C G1 1/6
C G1 G2 1/6
C G2 G1 1/6

In each of the first four cases, there is only one door the host can open. So each of these events occurs 1/6th of the time. Each of the last two cases, the host has two possible options - so each of these possible options only occurs 1/12 of the time. So the proportion of successes is 4 x 1/6 (=2/3) if you change your choice, 4 x 1/12 (=1/3) if you stay put.

I hope that this is now clear.

Fyz

hi fyz

the blurr point in ur argument is still where u assume the 1/12. how did u get that.

the challenge is what can we do to maximize the probability, which means after the first action (choosing first time) and after second action (which is removing 1 door out of the three)

u should count ur possible cases after the host revealed one door, all of the possible cases which are up to 8 now (4 switch and loose, 4 switch and win...)

shad

In order to get equal frequency of occurrence for all your versions, the host needs to open two doors for each time you have selected the door with the car. Then you would see two goats, and have 100% information - but it would still only be a single initial selection.

Another way of looking at this is that the host had the option of opening the door with the car behind it - that gives a total of twelve equal possibilities. When he sees the car, he swaps to the other one - effectively doubling the probability of that particular selection of door-with-goat

The basic principle is that a subsequent action cannot change the the frequency of occurrence of a pre-existing situation. If an existing situation occurs one time in 6, and it then splits two ways, the combined frequency of their occurrences can only be one time in six. In this case, we don't know that they would be equal, so it was an oversimplification to say 1/12; but whatever they are, the sum of their occurrence frequencies must equal one occurrence in every six. Simple logic - and also standard conditional statistics (though I've tried to avoid standard terminology, because it feeds so readily into the bad effects of poor early teaching).

If your early conditioning is so strong that the logic doesn't get through, the only way to get back to reality will be to perform a relevant experiment for yourself. But be careful that you don't double count when the host has two options - in each trial, he can only take one of them.

Fyz

fyz

somehow i can understand the logic you'r approaching the problem with.

what i can not understand and ask you to further elaborate it is the relation between poor early teaching, standard terminology and early conditioning...

BTW, i am working on the experiment thing you advised.. I managed to put my car behind a door and asked a friend to be a host. I still need two goats, know of any?

:D

cheers

If you find the goats, send them round when you finish the experiment. If you've got some suitable electric fencing, that would be good too. In the meantime, you could try substituting draughts pieces.

Have you considered the little difficulty of putting up with a game-so host for long enough to conduct a valid experiment?

Fyz

little difficulty - long enough - valid experiment

im suddenly concerned about this so called experiment. did u keep a log for such an experiment that ud share with the rest of the "poorly educated" as u like to call it?

im mostly concerned about ur declared need for the two goats when i'm finished... ur not willing to do the experiment urself for real no?...

shad

I don't have three large enough doors, nor do I have a sufficient number of randomised helpers. Also, I'd probably kill the game host and feed him to the foxes long before reaching a valid statistical conclusion.

In addition, I'm perfectly prepared to accept the validity of scaled experiments, for example using checkers.

I'm sorry if I appeared condescending - we all (I include myself) have mental blocks, and there are good reasons why this applies to statistics in so many cases. The problem is that in many regimes statistics has effectively been taught by rote in order to get students through limited specific tests. Because this is typically at a stage where students are impressionable, it can leave incorrect preconceptions that are hard to shift. Having read about Monty Hall since the start of this thread, it seems that one of the good things about the controversy was that it introduced experiments into the classroom at a very early stage.

Fyz

Fyz? - Your posts are becoming more surreal by the minute - too much chat with Kris, perhaps? Where the foxes did the new pets come from?? (I could have understood it if there were gremlins in the works)

True - but hopefully there is no chance of confusing my whimsies with my stuffy old technical stuff.

The foxes come from around my garden. They've been doing quite well since we banned the game-show host from the local hunt (it was the hot air that did the damage).,

Fyz

And I heard that too.

Quote:

Not at first well-received by Parliament, perhaps due to his dizzy theatricality of style, he ended his maiden speech prophetically: "...though I sit down now, the day will come when you will hear me."

Unquote:

There is purpose in my madness ( and possibly vv )

I heard that !

hello fyz

i admire ur "loyalty" to the scientific spirit and how u keep focused on the subject and goal...

i agree mental blocks do exist, and reasons are multiple.. even when sometimes absurd logic is not what happens in real life.

if i have this mental block on this case, why should so much people have it too, including the game show producers... shouldn't someone have told them that the host actions would provoke the player to switch and win the car 66.67% of the time...

i didn't see producers who would be happy to give away 66.67% of the time... of course unless they are USAID...

shad

Actually, the producers are happy to give away whatever number of cars brings them the largest numbers of viewers, and thus the highest ad revenue. A car is a tiny fraction of the cost of producing any weekly television show of any sort, where a single ad minute can be worth $1,000,000 to the network. Over time, the producers will tweak the probabilities to fit the business model, as necessary, but for many shows, giving away a car every week is pocket change in comparison to ad revenue. People like to watch people win big, thus the attraction of "So You Want to be a Millionaire". Lots of prizes = lots of viewers = lots of ad revenue.

In Monty's own words (paraphrased) "If I always offer a second chance, then it is definitely in the contestants' best interests to switch."

Except that reportedly Monty said rather different things on different occasions, and no-one said this was Monty's show (it is the tension or other aspects that sell the show - too high a winning rate would be as bad as too low)

Why should so many people have trouble with this statistical problem? I think that any serious statistics require a rigorous literalism that is actually counterproductive in most activities. That means either that it has to be something you can switch on and off, or that the capability will be incompatible with even the most routine of inventive tasks. Good engineers are more likely to be lateral- than literal-minded, whatever the media may say about you. So, axiomatic (pure) mathematicians excepted, engineers and other generalists probably need all the help we can get to develop these skills - until they become second nature. This has had to be part of my armoury for very many years, and I've been subject to some pretty fierce reviews - so I should have learnt something about it by now. Plus, whenever I am the least unsure, I draw splitting diagrams* (a bit like data bases, but with a proportion assigned to each branch); these are this insecure part-time statistician's equivalent of the experiments that I'm proposing for those who have not developed the statistical tools that match their personal approaches to problem-solving.

Sorry not to be more helpful

Fyz

Another possibility, if Fyz's experiments don't convince you, is to use a simulator. This link is to a particularly nice one, I think.

Because each possible outcome does not have an equal chance of occuring. For example, suppose I was to bet on a sports game. If odds were only based on all possible outcomes, the odds of either team winning would always be 50% because there are only two possible outcomes (assuming no ties, etc.). Your description should be as follows, realizing that "u" does not know which door is A, B or C:

1/3 chance u picked A, then host opens B
1/3 chance u picked B, then host opens A
1/3 chance u picked C, then 1/2 of this time host opens A
or 1/2 of this time host opens B

Shad,

Howetwo is right in stating that the choices you are limited to in the game are one attempt only at either A, B or C. You don't get two goes at this game. A bit like life really! The important factor is whether you choose to stick or swap once the other door is opened, not which door the host reveals a goat behind.

Sorry if I came across as over confident using "CAPS" but I was using them to add stress to certain words. I tried italic but didn't like the way it appeared on screen. I fear I was also the victim of too much wine, provoking a certain pomposity for which I apologise.

Fyz - The bit about my daughter was just allegorical. We share a similar sense of humour. I count myself truly blessed that she has also inherited my logical approach to problems, a love of mathematics, as well as a keen eye for art and ability to dance from her mother.. thank goodness it wasn't the other way around!

I was in no way trying too demean art either. One of my daughters' favourite places to visit is the Tate Modern; she has a Lichtenstein hanging above her bed. We really must return it one day!

Alun H

Alun H,

No need for apology man it is only a forum where we're all trying to show the right and prove the wrong ones they're wrong - for some of the people here -

sometimes u see here and there teasing or tingling comments on posts just to pump up the action and spice up the debate..

in anycase, goat or car, u shouldnt drive while drinking, so better stick with the host choice no? :D

cheers.

sometimes u see here and there teasing or tingling comments on posts just to pump up the action and spice up the debate

That is probably an understatement !

That is probably an understatement !

an anonymous post is quite the proof to what i just said

shad.

I RESENT WHAT YOU SAY ABOUT ANONYMOUS POSTERS!!! I THINK YOU ARE A STINKY HEAD!!! AND IF I HAD MORE TIME TO SIGN IN I'D SAY THIS UNDER MY USUAL HANDLE!!!!!!!

A NON-MOUSE

Hi there Ratty

Does the recent preponderance of this style of post mean that the aspects relevant to the nominal thread are now fully chewed up, digested, and converted to hot air?

Mole

yeah right... whatever

i agree if u had more time to sign in as it would take so much time for a slow head like you to put ur name in.. u could at least post anonymously and put ur name on it unless u r shy of it...

the only stinky thing is ur low post so keep it up, there must be someone proud of u somewhere in ur twisted mind...

shad

As written, it only describes a single occurrence. I think it was intended to mean what you say (and what we calculated), but it doesn't specifically say that, unfortunately

Fyz.

The CR4 folks took out a reference to the host always showing a goat, and always picking a door other than your first choice. I think that their edits improved the question, (maybe dramatically) because it better represents probabilites in the real world. Often, you don't know what the history is, and as we do here, need to work only with what can be observed at the instant.

Suppose I went onto such a game show. I would have no idea how the show is run, what the host's motives might be, what the history has been, etc. I simply don't watch enought TV (especially game shows) to be able to discern any of that. All I know is I've made a guess, the host has shown me the location of a goat (I assume because he wants to help me win -- but I could assume he is trying to confuse me, or build suspense, too), and has said that I can change my guess, if I want. From the instant data alone, I have to make my decision. And from that data alone I can calculate probabilities and do so.

Although from a great many observations we can determine the probablity of some occurence in the future, we often make on-the-spot probablity assessments, based only on the instant data. Gamblers would do better if they relied more on instant observable data, and less on history, (or their perception of a too-short segment of history). The Bayesian approach, I think, requires great care. Gamblers say "It's landed on red ten times in a row... it's got to be black next!" Bayesians say "It's landed on red ten times in a row... it's got to be red next" Mathematicians say: "Half the spaces are red, half are black. The chance that it lands on black is 50%." (This is, admittedly, a "glittering generality", as my english teacher use to say. )

You're the contestant. Remember, the objective is to walk out with a car. To you, at this instant, it doesn't matter what the host has done in the past, because whatever he does, he's already done it to you. You only have one choice to make: stay or switch.

You can, as you have done, calculate the probabilities here while making only a few assumptions (beyond the question data), most of which are so obvious that they don't need to be stated (no, the goats don't wander around, yes, there really are two goats, one behind each of two doors, etc.) What the host has (reasonably) done in the past has no effect on your current situation. The audience is waiting...

I prefer the CR4 version to my own. If I were going to edit your post, I'd write: ".... doesn't specifically say that, fortunately."

Sorry for the long post. If I had more time I could make it shorter.

P.S. I think you are unfair to your daughter - some test questions are unclear, and most life questions are. Experimentation (my game) is just one way physicists ask supplementary questions to clarify the initial uncertainty. Much of art is about questions of human perception and behaviour - and often clarifies or highlights uncertainties here.

Questions always need to be clear ;

A mathematician, a physicist, and an engineer were traveling through Scotland when they saw a black sheep through the window of the train.
"Aha," says the engineer, "I see that Scottish sheep are black."
"Hmm," says the physicist, "You mean that some Scottish sheep are black."
"No," says the mathematician, "All we know is that there is at least one sheep in Scotland, and that at least one side of that one sheep is black!"

As you will see , no clear question was asked. Ooops. Queue internecine war

Your valiant search for clarifications will win the day.

Perhaps they were all wrong - the window was dirty, and they were looking at an Aberdeen Angus that was further away than they realised.

... or a fly on the window....

Very kind of you to ignore my messy mistake with cue/queue Fyz . Gracious as ever.

What intrigues me here , is that I have not noticed a post by anyone who has taken a few minutes to write a simple computer program for themselves. A room full of arguing Engineers , and not one does the obvious . Perhaps they might find that people would say their program as wrong. Second hand programs are not worth a banana-skin.

"Queue" produced a beautiful(?) image of engineers standing in line awaiting opportunities to indulge in ethnic cleansing.

On which topic, why do the euphemisms so often become the accepted descriptions - "ethnic cleansing" is usually genocide, paedophilia is pederasty...

There is even one function for which I have failed to find a formal descriptor that was not originally a euphemism. In approximate order of first occurrence:
Jakes (same as Jacques, or John), privy, head, closet, lavatory, toilet...
Water closet is a possible exception; although it is based on one of the euphemisms, it does at least have no alternative meaning.

P.S.1 Definition: Scatology - the study of absent mindedness
P.S.2 New name for WC - SCAT socket?

Try road/trolly/air..... rage for other stupid and foolish headline grabbers.

'Jakes' would have had it's precursors. The said article was probably one of the earlist things to be euphemized.

P.S.1 Definition: Scatology - the study of absent mindedness

You miss the other definitions of this (or do you )

'Going to engage the SCAT socket' sounds suitably Engineer-speak , although 'going for a dump' will probably persist on site.

Missing definitions - I've only stated one, but I know several more that I find less amusing - and there are almost certainly some that I don't know (spare us unless they are really good)

Fyz

Hello

As is stated in the problem we can deduct that the host always select a diferent door to the one the contestant selects, ( let this condition be called "TH", if this is the condition we have the following possibilities:

1. If the contestant selects the door with the car, host will have two possible choises, the goat A or the goat B doesn´t matter and the contestant should change his selection because the host will act according to the condition TH. In this way the contestant will get the car.

2. If the contestant selects a door with a goat and the host opens a door with a goat it means that the remaining door has the car, so according to the condition TH the contestant should stay in the selected door. In this way the contestant will get the car.

Now, we have two posibilities with different recomended action:

Possibility 1 we recomend to change

Possibility 2 we recomend to stay

Now we know that we have 33% of posibilities to be in the case 1 and 66% posibilities in the case 2, so our recomendation is to stay because in this way there is a better chance to get the car.

Sorry for my english, my mother language is spanish, I hope I make an understable explanation.

Regards

Ciro

You had a 1 in 3 chance to win the car until the host opened a door that revealed a goat. Now you have a 1 in 2 chance of winning the car, so it doesn't make any difference if you stay with your original choise or if you choose the other door - each of the remaining doors has a 1 in 2 chance of being the car.

Bill Koser

Barron, Wisconsin

You actually take for granted that the first door that host opens is a goat so the Ω=(c-g2,g2-c,c-g1,g1-c)....if you call A the possibility of being the car behind the door you have picked A=(C-g2,C-g1) P(A)=N(A)/N(Ω)=1/2=50%..you change your mind about the door.You don't change what is behind the doors..if you pick the other door the Ω still remains the same but now you want the possibility of not being the car behind your door .This is called P'(A) => P'(A)=1 - P'(A)=1 -1/2=50%.. this problem can be faced with maths that I have been tought at the age of 13..it's a very simple math problem....(I want to apologise for my English,they are not so good as they used to be...I hope that you understand what I'm writting)

I know this is not the math answer that you may have wanted to hear, but it seems that this is more of a psychology question. It might be considered a statistical question if the following is true, but that is only a description of the issue not a deciding factor. If the show or host has a history of steering the contestants in the wrong direction then you should stick with the original pick. You definitely don't want to change if he only sometimes elimimates one of the picks. If, instead, he tends toward giving away the prizes then you should change your pick. It is clearly a fifty-fifty chance if there are two doors, since he eliminated one of the three for you.

you can also use thermal googles to detect heat behind the doors....

if that's what on your mind, i recommend using x-ray goggles. you'd see a naked goat!! how about that. :D

cheers.

The confusion found around the world regarding this question (which is known as the Monty Hall paradox) is mainly psychological. In other words, people will "imagine into" the situation all sorts of things that don't really matter in determining probabilities for the current contestant. The problem, as stated here, is much like determining the probability of a fair coin flip, in this sense: there is a definite and clear calculable probability for one path or the other, that does not depend upon history or motive. In a coin toss, you can say with certainty that the probability of tails is 50%. But we know that it is quite possible to have 10 perfectly fair coin tosses in a row, with heads every time. That does not suggest that the calculated probability is wrong.

In Bayesian logic, the history of previous events is used to determine the probability of future events. So, with the coin toss scenario just described, the Bayesian logician would come up with something other than 50% probability. (I'm short changing Bayesian logic here -- but I think you'll get the drift.) But we know that a coin can end up tails many times, and the probability of the next throw is still 50% for heads. The coin has no memory. (Obviously were assuming fair coins, fair tosses, etc.) Bayesian logic does not apply here, because you don't have access to game show history. (That's an oversimplification.)

Here (back to the game show), all we know is what is described in the question. We could assume that "you" are the only contestant ever, and that there will be none in the future. The probability does not change.

So... we'll call you the contestant.

You've already picked a door. The host opens a different door, and shows you a goat. Why did he do that? It doesn't matter. If he did it to be mean or kind does not change the probabilities. Did you pick the car on your first guess? We can say, with mathematical, probabilistic certainty, that before you do anything else, you have a 33% chance of having selected the right door to win the car. So do you stay with your first choice?

Turns out, you have only two possibilities, right now: stay or switch. So if the probability of the first (staying) is 33%, then the probability of the other must be 67%. In one sentence, that is the answer, and a reasonable explanation, I think. This is the explanation I like best.

But we can go further. What is the probability that your first pick is a goat? 67%. No one will argue this. Therefore, when the host showed you the "other" goat, he has increased your chance of winning even beyond 50%. Now you can be 67% certain that the car is behind the remaining door. (In other words, in those 67% of the times when you picked a goat, and he has shown you the other goat, you can be 100% certain that the car is behind the remaining door.)

For yet another perspective you (as the contestant) could say: "Well, there is a low probability that my first choice is right. There is no chance at all that the door opened by the host is right, I can see that. Therefore there is a high probability that the remaining door hides the car.

The 50/50 choice is an illusion*, because it ignores the fact that the host has shown you one of the "wrong" doors, improving your chances of picking the right door, if you have your wits about you.

Whether or not the host acts in consistent or predictable ways has no effect on your contest: he's already acted by the time you need to make your choice to switch or not.

* Unless you make it applicable by randomly choosing to switch or not.

Ken

You are pushing it - conditional probability does depend on the conditions. The extreme example is where host filters his actions, and only exposes a goat when there is a car behind your door. If you have history to go by, you can observe that this probably is his behaviour, with confidence increasing according to the number of occurrences observed. Your additional uncertainty is whether host might change behaviour as you appear on the scene. That is to say, unlike your initial uninformed selection of 1 of 3, host's selection is not necessarily a fair coin flip as you are assuming, and history will allow you to assess the bias.

It is also untrue that random selection by host gives the 1/3, 2/3 result. There are four equal-probability cases where host selects the goat. Writing in the order of your selection, host's selection, unselected, these are
C G1 G2
C G2 G1
G1 G2 C
G2 G1 C
Clearly, if host is selecting at random (fair coin host), the probability is now 50% whichever way you choose (two of the cases where you might have changed your mind to the car have been excluded from the set).

Fyz

Hi Fyz.

Pushing it??!!

After all this work to win the car, I have to push it? Don't the show producers have the decency to at least fill it with petrol?

host's selection is not necessarily a fair coin flip as you are assuming

I agree with everything in the paragraph from which this was extracted... except for this quoted segment. I'm not assuming that the outcome a fair coin flip determines the host's selection of a door to open. I brought up the coin toss only as an analogy to indicate that this is not an exercise in fuzzy logic. In other words, you can calculate hard numbers for the probabilities, provided you specify the assumptions with which you are working.

I think you can arrive at reasonable assumptions without them being stated in the question itself. In arriving at reasonable assumptions, you could consider various possibilities. For example, there might be times when the host shows you what's behind the door you chose. There might be times when he shows you the car, regardless of your choice. Once in a while, he might operate on a different algorithm, and show you a goat, only if your choice is the car. (Obviously he could not do that for long, because contestants would catch on and win too many cars.) But if you look at the effects of the various assumptions that you might decide to work with, you would probably come up with the assumptions under which the paradox is usually proposed. These assumptions are, I think, a pretty accurate reflection of the way in which our hero, Monty, actually behaved. One would think, (hope) that people would then simply always switch -- that they'd catch on. I fear that is an optimistic view -- especially because game shows usually screen contestants for their ability to act wildly emotional and irrational -- it makes for better TV.

But maybe Monty threw in an occasional other behaviour. Dunno. I'm guessing you'd need to know quite a bit of the show's history to determine if Monty was operating under something significantly different than the assumptions usually presented in descriptions of "The Monty Hall Paradox".

So, while it would be nice to know some history, for the question as stated by CR4, you don't have that history, so you have to "make it up" in the form of stated assumptions.

Incidentally, when I submitted the question, I asked that they not call it "The Monty Hall Paradox" knowing that that would then lead to many web searches, which would probably make the discussion less interesting.

Oh... here's another thought*. When I say that history does not matter in the instant, that's because the contestant does not have access to that history, as the question is presented here (i.e., not as it is usually presented as the Monty Hall paradox.) So here, the situation is no different than your coming up to a street performer who presents you with some logically equivalent variation of this. You've never seen his shtick before, but you can still conclude that by showing you a "goat", and offering you a chance to change your guess, he has improved your chance of winning. Knowing nothing else, you know that 1/3 is the chance that your original guess wins. For you, in the instant, there is only one other alternative. The other alternative must then have a 2/3 probability of winning. Do the previous two sentences make sense? (I'm beginning to waiver here.)

(If you had some history then you could further improve your odds beyond 2/3 -- but then the logic becomes a little fuzzier -- because you might want to filter out behaviours that were clearly flukes -- you think -- just as you do in experiments. And the host has probably learned to either be more consistent or less consistent...) (Of course, flukes in experiments have a precise definition: they are the things the don't support your hypothesis. )

Sorry that I am probably repeating things I've stated elsewhere.

* And this mainly just restates my drivel in other words.

The only way you can do better than 2/3 is if the host helps even further than always opening a door other than yours with the goat behind it - for example sometimes opening your door if you've selected correctly.

Then, if you don't know what the host is doing, you can only state what would be the best strategy under specific assumptions - and you mischievously appear to imply that all reasonable assumptions must lead to the same answer. If you seriously believe this, I'd say that it's the goat you should be pushing - and it would be pretty aggressive if 'they' had filled it with petrol.

If host only shows you the goat if your choice is the car, that wouldn't help you at all - unless he always did that, and invited you so switch whether or not he showed you a goat. In that case the best decisions again give a 2/3 overall chance of winning.

The only reasonable assumption I know would be that a major requirement for a games show is to raise money, and they do this either by advertising revenue or because viewers are willing to pay. That means they usually use "names" to attract initial viewers. Then the show must be attractive to watch - that means you need some tension, so you can't fiddle it so the viewer always wins. Now, the revenue increases as the show becomes successful. So do the number of contestants - and most contestants are regular viewers (and those who aren't are probably there for the winnings, so many will study the track-record anyway). So your statement that it is no different to a street performer coming up to you... is most unlikely to be correct (and this is cast as a probability+ question). If it was a street performer, I'd be pretty certain that it would have to be weighted against my winning - at least in Europe (where would these guys get the money to give away anything significant?)

Now, returning to a games show with a consistent strategy that looks as if there is some variety: a neat strategy for excitement might be that the host guesses a second door at random when you have guessed yours, and is only then told through his headphones where the car is. If he guessed wrong, he will open his door and invite you to switch. If he guessed right, sometimes he will open his door, and sometimes yours. Under these conditions it makes no difference whether you switch or not - but it adds to the tension and the contestant's feeling of control. That's a cheaper strategy for the game-show than always opening a goated door to give you the choice, as you win 1/3 of the time whatever you do.

Now I've read up a little, I understand that this was just one of the games on the MH show, and that a large part of the show's "attraction" was that MH "played on the psychology of the contestants". So in that case, his behaviour would depend on the contestant and on MH's mood (what else would you expect from a games show host with a B.Sc with LLD?). In interview, Monty Hall stated that Marilyn vos Savant's statistical assumptions in no way resembled what happened on his show.

Returning to the basics - there are too many reasonable assumptions you could make that all have different outcomes. I've covered a few of these (by no means all) in various postings. Personally, unless I had some history or some other insight into the host's behaviour, I would not switch - on the basis that I maintain my 33.3...% probability by not switching, and only one of the more probable systematic strategies that exposes a goat is actually of benefit if I do switch (also because another of them reduces my chances to zero if I do switch, and I have no reason to suppose the strategy is consistent or even helpful from show to show)

Fyz.

You miserable cheapskate! Here I am trying to give away cars, and you are trying to keep them!!

I knew I'd eventually get your wall of clear logic and reasonableness to break down if I spewed enough mischief and ambiguity!

• Fyz says: I would not switch -- on the basis that I maintain my 33.3... %. And all along you been saying that you would switch!

Of course I'm kidding. (Imagine how frustrating it must be for politicians to get quoted out of context.)

I agree with virtual everything you say here (and elsewhere.) I also keep laughing when rereading the sentence with the petrol-filled goat. It's quite an image!

There is a slight difference in our approaches to the question as reflected in this:

Then, if you don't know what the host is doing, you can only state what would be the best strategy under specific assumptions - and you mischievously appear to imply that all reasonable assumptions must lead to the same answer.

I agree with the first half of that completely. But regarding the second half: what I intended to imply was that, in answering a puzzle question (and especially a puzzle question like this if it were on a Probability 101 test), the simplest assumptions are probably the best. So having seen that the host responds to your guess by showing you a goat and offering you the chance to switch answers, you can infer that those are the "rules" of this game (possibly incorrectly, but you have nothing to indicate that). You could also infer that the contestant knows these rules, possibly because they were made explicit, or possibly because the contestant has seen the show before. Many other assumptions are reasonable as alternatives to this, but nothing in the question indicates that these alternate assumptions are any more valid -- and some are very unlikely as consistent behaviour. (For example, it would be unlikely that the host consistently shows a goat and allows you to switch answers only when you pick the car: obviously you would not switch.) So, without spending too much time, I'd state those assumptions and then give the stock 1/3 - 2/3 rationale. I think that Marylin was reasonable in making those assumptions the first time around, although she certainly should have made them explicit.

I'm essentially arguing in favor of giving the answer "the teacher" expects -- the one that might point out some interesting concept or clear up some misconception. Of course, I am not at all consistent in this philosophy -- in fact, I usually have a strong preference for being contrary. (Will the boat move -- Of course! Getting it to stand still would be the challenge, provided the drag figure is a typo.)

I had a really great physics teacher in high school and we got along very well. Had he presented me with the Barometer Question, I would have off-handedly said: "Well, let's see... we can easily eliminate measuring differences in air pressure as a possible answer, because a 1,000 foot building would only move the barometer needle 1/2" along its arc, so the resolution would be way too low. Let's see, what would some more realistic answers be..." He loved that sort of thing, and we had lots of fun.

I like your strategy for maintaining tension and excitement, while increasing the likelihood that the contestant goes home with a goat... you cheapskate!

Ken

On the other hand - if the host had no way of knowing where the car was, you'd always have more chance of getting the car if you switch. It all comes down to your perception (or knowledge if this format has been going a while in the show) of the way the show is attracting or holding its viewers. But you should never forget that I already have as many cars as I can possibly use, and a very large and verdant lawn...

Ken

Now that I've a minute, I'll give a bit of detail as to why I would not switch if I found myself on a game show.

First, if it had been in the 1970s, and I had inexplicably chosen to go on a US game show, I would have switched. That is because I would only have attended a "successful" game show that made lots of money. Assuming that the host didn't have visible headphones, the most probable situation back then would have been that the host had no idea where the car was, so switching would have made no difference to the chances. The next most probable situation would have been that host had a way of knowing, and wanted to help, and, whenever he made a selection, it was always a different door with a goat. In that case the chances of a car became 2/3 if you switch.

Now fast forward to 2007, and I have accidentally wandered into a UK games show. The market is much smaller, and advertising is deserting television in droves. The games show would have to have very few takers if they needed to wangle my attendance (SFIK I'm not a celebrity) - so it's a marginal exercise for them, even if it is possible these days to produce a game-show quite cheaply. Plus, it's an outsourced production company without a direct link to the advertising. It's easy to provide a totally hidden headphone, they are out to save every penny, and they can't do anything about the probabilities if you don't switch.

Fyz

Here is a nice general link that contains a good reworking of Shakesperian trickery.

Kris

Now I know who is the little terror that steals the bird seed.

Fyz

And you also know the one who Ken borrowed seed from (with my blessing). ()

For more current help , examine the word 'more' in #9 of current CQ

I think you can leave the unconverted here alone now that the 'Answer' is published. How on other anybody can't program this Q to see themselves is beyond me - at least that prevents suspicion that the nice demo program is rigged (or would it - some will never believe! ).Kris

And you also know the one who Ken borrowed seed from (with my blessing).

I am hoping that this treatise on ethical seed-sowing will convince my wife that I should begin at once to go forth and attempt multiplication -- at least in my corner of the universe.

suspicion that the nice demo program is rigged

Now that Microsoft has it's claws into everything worldwide, isn't anything that runs on a computer suspect?

Not at all. You can still get stripped-down versions of Linux Unix etc. that predate efforts at compatibility - but how much longer?

True. I've been running Windows for many years, mainly because my clients have had it. I have W2000 now and have no desire to replace it with Vista -- mainly because W2000 has been fairly stable (in a Microsoft wort of way) and there is nothing in Vista that I am dying to have, and I am unwilling to by the new hardware to fully support it.

Now that hard drives are so cheap, I think I'll put another on this machine, and use Linux on it, and boot whichever operating system suits my purposes.

"in a Microsoft wort of way"

Is that "wort" or "wart"? Let's look at the definitions:

wort

3 entries found.
Main Entry:

¹wort

Pronunciation: \ˈwərt, ˈwȯrt\ Function: noun Etymology: Middle English, from Old English wyrt root, herb, plant — more at root Date: before 12th century : plant; especially : an herbaceous plant — usually used in combination <lousewort>

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Hmmmm, I don't think a MS "plant", "herb", or "root" would make much sense here, although you have to wonder if some MS programmers were smoking "herb" at the time they wrote some of the software! How about:

wart

2 entries found.
Main Entry:

wart Pronunciation: \ˈwȯrt\ Function: noun Etymology: Middle English, from Old English wearte; akin to Old High German warza wart, Old Church Slavic vrědŭ injury Date: before 12th century

1 a: a horny projection on the skin usually of the extremities that is caused by any of several papillomaviruses —called also verruca vulgaris b: any of numerous similar skin lesions

2: an excrescence or protuberance resembling a true wart; especially : a glandular excrescence or hardened protuberance on a plant

3 a: one that suggests a wart especially in smallness, unpleasantness, or unattractiveness b: defect, imperfection — often used in the phrase warts and all

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Aha! I think the last meaning, 3b, defect or imperfection really describes MS products very well! We usually buy and use MS products, warts and all!

Hey - you missed 'wort' as in the semi-fermented stuff used in making BEER !

Any idea how it is that some posts here end up as 'Re :Fyz...' and others look like 'Re:whatever...' ? I'm baffled.

Yes, all you have to do is edit the title box when you write your message and keep the "Re:" part of the default title (which is usually the response to the title of the original Blog Entry).

Thanks STL . I never thought about that . Talk about missing the " bleedin' obvious " as Basil Fawlty would say.

Yes, I agree completely: I deviantly mend "wart."

Now that Microsoft has it's claws into everything worldwide, isn't anything that runs on a computer suspect?

I may carve those words of wisdom on my monitor. I have a liking for writing simple routines myself , on the basis that if I can't work out the logic I'll be stuffed the day I need to. I'm a pessimist with smile.

Anybody for a game of division with Intel ? - That was just the mistake kind of event. For some fun paranoia check out the thread on Canadian coins.

From a statistical viewpoint, it does not matter- there are 2 doors, one with a goat and one with a car. Your odds have improved to even from 2 to 1 against, but the doors are the same.

Now if you are familiar with the show, and know the host finds a goat when another goat door is chosen, then you should chose the other door.