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

If he always opens another door with a goat, you should still change your door. 1/3 of the time you had the correct door, and that will be wrong. But 2/3 of the time, you will have chosen the wrong door, and the host will have removed the possibility that you choose a wrong door if you have got it wrong.

Earlier in this thread (Post #139) I responded to "Physicist?" with a post that concluded with what I considered to be a humorous ribbing, but which was in fact taken as an insult by him, as well as "Blink", who found the need to come back at me as well, although it did not involve him at all. By the way, my use of the word "feces" was a carry over from my misspelling of the word "facetious" and another persons attempt to use it as a joke aimed at me.

I wonder just what others thought about this little exchange and whether your country of origin and social mores had anything to do with how you perceived it. Please limit your response to a choice of one of these:

1. Ha! That was very funny STL. I got a good laugh. @%&$%& 'em if they can't take a joke!

2. Hmm, not that funny, but not that insulting either. They really should lighten up.

3. Mildly insulting, you should lay off that kind of unfunny, crude humor, STL. Obviously, though, you did not realize you were being insulting.

4. Over the line, STL. You should know better, and in fact, I think you might have.

5. Way, way, over the line STL. You need to go back to Remedial Manners 101 and don't be so immature and aggressive in your responses. You were, and intended to be, highly insulting, and not kidding at all.

So please, pick a number from above (1-5) that represents your opinion of my response to "Physicist?" at the end of post #139. Please also state your country of origin for reference. I appreciate any and all constructive, non-insulting feedback.

No "Guest" comments please, only legitimate, logged-in, CR4 members.

Oh, there is option number six:

"Stop it, stop it now, all of you! Quit wasting our time and your time with this silliness!"

7 - This site would die a dusty death without humour . Any humour will be found distinctly non-funny if over analysed. A paradox indeed. On a Global site , humour is essential to cope with cultural/linguistic difference. On the rare occasion that it seems to go belly-up , people explain their reasoning and move on . This has happened . I have no doubt that STL and Fyz are both great people , who's comments I always enjoy reading. Next communication between the 2 should start with a smiley and the phrase "We will get along in future - history is history"

Failure to do so may necessitate ruthless lampooning of said parties. " We will get along in future - history is history"

Subtle variations will be accepted

Let's see.....if the host shows a goat, does he become a goater? If so, does that make the contestant a goatee?

I crack myself up!

Isn't the term "goat-herd" - or is that a combination of your accent and my encroaching deafness? And have you ever tried goat tea - to my mind, would be better if it tasted fowl?

Is he a carer when he shows the car? And, if so, are the contestants carries or caries?

I must be cracking up to go on like this.

Fyz

Your humour is starting to appear .

Your reference seems to be to filling cavities, not caries

Unlike some of the good folk of St. Louis my humour does not extend to crack . I would assume it does not to yours or STL's either. A bit like making anoerexic jokes in Biafra. Those are the nice pics. Humour is in the eye of the beholder perhaps.

Please forgive me if this has been addressed in the hundred plus responses, but the original question did not tell us whether the contestant revealed his first choice to the host; it merely says "The contestant picks one door ...". If not, then the host's choice is independent of the contestant's choice and clearly not influenced by it. In addition, we are not told whether the host knows which door conceals the car; again, if not, his choice is random.

Either (or both) of those "ignorances" makes a big difference, because it removes the psychology, and leaves the contestant with a 50% chance of choosing the correct unopened door.

If the contestant does reveal his choice, then I don't think it is possible to say whether it is better to stick to the initial choice or change.

Ken G

None of us will get the car......So whatever we say is meaningless..

But it's the goat I need right now...

You'd have to change doors given the second chance.

In the first round you have 1/3 chance of getting the right door. This allows for a 2/3 chance aginst you.

Having a second chance to switch doors, remembering that you were probably wrong the first time, and 1 of the 2 remaining doors has been removed, the other door not yet selected would have the car.

That is just too simplistic. You don't know your original choice was wrong. What if the host only ever shows you another door if you chose right the first time?

Good points. In answering, you'd need to state your assumptions.

I think the simplest assumption, given no other information, would be that the host always opens a door with a goat (different than your guess), and always offers the option to switch. (In other words, that the sequence described in the question repeats.)

For consistent patterns, you could reasonably rule out his only showing a goat when you pick correctly -- because he would then be identifying the car location. Likewise, you could rule out his only showing the other goat when you pick a goat -- because that would also indicate the position of the car. If he always did one or the other of these, then he would always be showing you a goat, which gets back to the simple assumption.

But there are certainly other possibilities, and one would need to specify the answer assumptions accordingly.

The answer given (that changing is a better option) is both logical and counter-intuitive given the assumptions made. However it reminds me of a card trick that involves taking away the incorrect choices independently of the choice of the "volunteer" resulting eventually with the revelation of the correct card.

Consider this possibility - if the aim was not to stretch out the show for as long as possible and you selected a door that concealed a goat the logical thing would be for that door to be opened leaving the contestant with the goat as a booby-prize. If the door contained the car another door would be opened letting the contestant have another attempt at getting it wrong. Of course this behaviour would get predictable after a while. If this was a regular part of the show I would say switch (although it makes my head hurt to assign 66.66% probability to this being the right decision). However if it was a standalone event I would say stick with your first answer.

You should change de door you have choosen, because the first time you had a probability of 0.33, and now you have to choose between only 2 doors and that`s a probability of 0.5.

Here`s a practical demostration:

Let´s assume that the price is in the first door.

First scenario: you choose the First door and then the host opens any other door:

if you change: you LOOSE

if you don´t change: you WIN

Second scenario: you choose the second door and then the host opens the third door:

if you change: you WIN

if you don´t change: you LOOSE

Third scenario: you choose the Third door and then the host opens the second door:

if you change: you WIN

if you don´t change: you LOOSE

As we can see, changing the door increases the chances of wining

Pablo H.

You all have it wrong the host reveals a goat, this door is taken out of the equation the people behind the scenes swap the car to behind this door so witch ever other door you chose you will always get a goat. The result is fixed, when you cry foul the car is where you had not looked.

I read the answer, ran the simulation and I still believe the odds are 50% of winning the car. Given the condition that the host will always reveal a goat and provide an opportunity to make another choice means that the first choice is totally meaningless. The first choice has no effect on the second choice which is between two doors. Say you initially chose door 1 and Monty revealed a goat behind door 3. Not switching = choosing door 1. Switching = choosing door 2. It is a choice between two doors with a car behind one and a goat behind the other. The fact that the simulator gives a result that favors switching only proves that the algorithm used in the simulator is incorrect.

You probably agree that the probability of picking a goat on your first guess is 2/3. Because Monty must show another goat, that means that 2/3 of the time, the remaining door hides the car. So 2/3 of the time, if you switch, you win.

If still unconvinced, read the Wikipedia link (in the answer above), which approaches the problem from several angles. Then step through the simulation with the "cheat" box checked, so you can see exactly what is happening. You can do through as many single trials as you like, and the probabilities will get closer and closer to 2/3 and 1/3.

Finally, if still unconvinced, you can write your own program as a simulator.

GIVE IT UP KEN !!!!!! YOU ARE COMPLETELY WRONG AND YOU KNOW IT!!!

What's got into you Kris? Do you want to be banged up for disturbing the peace? Ken's stated assumption is that host is following an algorithm - he deliberately opens an unchosen door that hid a goat, irrespective of the success or otherwise of your initial choice. On that basis, the 2/3 statistics are correct, and, dare I say it, so blindingly obvious that only extreme prejudice could prevent you seeing it. So read very carefully, I vill write zis only vonce (here):

On 2/3 of occasions, you will choose a goat initially. If that happens, host will reveal the other goat. That means the car must be behind the remaining door. So switching gets you the car. Repeat - this covers 2/3 of occasions. So switching gets you the car on 2/3 of occasions (under these assumptions about the host's behaviour).

If you can find a hole in this logic, be specific. If not, be as big as STL Engineer...

Fyz

P.S. BTW, this sort of thing can be due to finger trouble and/or miscommunication, and is one of many reasons that I mistrust simulation results unless/until the simulator has been extensively reviewed, proved against practice, and/or independently duplicated. This applies even when (heaven forfend) I have written the simulator myself.

If you read carefuly you will see that I do understand the problem, and have programmed a simulater Fyz.

"If not, be as big as STL Engineer..."

Hey, that's not nice. I am working on my weight problem!

( )

Apologies - had I none their mite be sensitivity, I should of bin moor car-full of de warding. (Just like what I've dun with the spill chucker her).

Fat Fyz

!!!NOT MERELY WRONG, BUT WRONG-HEADED!!! (IF ONLY WE HAD COLOR!)

!ESREVER NI ETIRW EM EKAM T'NOD!

In the first round of choices you have a 1 in 3 or 33% chance of success. In the second round you COULD have a 1 in 2 or 50% chance. If you do not choose to change your choice you are essentially still playing with the first round odds of 1 in 3. 1 in 3 are loosing odds and the stats will play this out. When Monty opens the first door the game really starts again with only 2 doors and its simply a 50/50 chance. Your chances are better in the second round as long as you treat it as a whole new game.

If you rechoose at random in the second round, you obviously have a 50% chance. If you stick or switch, the probabilities depend on host's algorithm. For the algorithm stated in the solution*, the chances are 2/3 if you switch. Here's the simplest and most compelling explanation that I know:

On 2/3 of occasions, you will choose a goat initially. If that happens, host will reveal the other goat. That means the car must be behind the remaining door. So switching gets you the car. Repeat - this covers 2/3 of occasions. So switching gets you the car on 2/3 of occasions (under these assumptions about the host's behaviour).

Show the error in detail or accept the inevitability of the logic...

Fyz

*Host opens an unchosen door that he knows hides a goat regardless of whether your initial choice was successful.

So, it is important to state ones assumptions is it? What other assumptions could have been stated? I think some fairly fancy and contrived ones involving thermonuclear war or that the goat is a car and the car is a goat. Or perhaps something more subtle: Monty has no idea where the car is and just got lucky with the opening of the first door.

Knowing the answer to the question, which contains much of the information relevant to the question (call them assumptions or call them post justification) leads to some rather varied responses. The discussion may well go on . . .

Come on okes... you are complicating the issue. The contestant chose a door. The host selected another door and opened it revealing a goat. The host has set a trend. No matter which door the contestant choses next, the host will open the other. So, is it not true that the host knows exactly where the car is, but he also needs to be consistent. He'll have to open the door not chosen by the contestant. For the contestant to win the car he'll have to switch and chose the other door in order for the host to open the original door selected by the contestant. The fact that the host didn't open the first door is because he knows the car is behind it. If the contestant picked this up and knew the host has to be consistent, the car is definitely behind the first door. Therefore increasing his chances of winning to 100%.

Just a thought from someone who is not mathematically orientated.

Your version is the complicated one - and totally misunderstands the dynamics of game shows. Whatever order host opens the door in, the contestant gets what is located behind the door of the contestant's choice. The choice is whatever the contestant most recently said it was.

Fyz

From the web link in the posted "official" answer:

"This difference can be demonstrated by contrasting the original problem with a variation that appeared in vos Savant's column in November 2006. In this version, Monty Hall forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied, "If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch" (vos Savant, 2006).""In this version of the puzzle, the player has an equal chance of winning whether switching or not. There are six possible sequences of events that can occur, each with probability 1/6."

The Challenge Question does not state that the host knows where the car is or that he always opens a door with a goat because he knows. All the question says is that he opens a different door showing a goat. So, the question is ill defined to say one way or the other without adding assumptions to the problem - either assume the host knows and always chooses a goat or he is clueless and has the possiblity of opening a door with a car, the answer is different for each case.

If you assume he is clueless, the question doesn't indicate he isn't, it doesn't matter. If you assume he always opens a goat door, it does matter - you should switch.

If only those were the only possibilities, there would be no risk in always switching. But a less beneficent host (2007 economics) could choose only to open another door and offer a switch if you selected the car initially. Of course, if you knew what was going on, you'd know to stay put.

What if I want a goat, not a car?

Cars pollute and contribute to global warming. But then again, so do goats...

Assuming that it is roadworthy, you could take the car and sell it, and buy the goat and have enough left over for several years' vaccinations and feed. (Depending on breed, goats typically cost up to $300.) If you'd rather take one home from the show - negotiate.

When you do a decision tree, it is clear that if you have chosen the goat, you must switch to win; if you have chosen the car, it pays to hold- since there are 2 out of three chances that you have chosen the goat, it pays to switch. It is non intuitive, but very clear once you draw the diagram.

There are 3 doors, contents unknown; you have one-in-three chances of selecting the winning door.

Once the goat is revealed behind one of the three doors, you are left with 2 doors, contents unknown. Whether you pick door 1 or door 2, it makes no difference on your chances (50-50 probability which one holds the prize). The fact that you previously picked door 1 or door 2 makes no difference because the result of your initial selection was not revealed!

You have a choice between opening the unknown door that you had previously picked, and the unknown door that you had not previously picked. The key words are "unknown door"; shame on you, if you think there is new information in re-confirming or switching your choice.

Is this a semantic argument that "unknown" must mean unknown to everyone? Someone placed the goats and the car, so that cannot be the case. The interpretation that "unknown" merely means unknown by you the contestant is therefore as valid as any other. That allows the host to have knowledge, and to make an informed decision as to which door he opens.

If, on the other hand, you are saying that you cannot deduce anything from the host's choice, consider the situation if he chooses to open the door that conceals the car. Then re-analyse what happens with the procedure that is suggested in the "official" answer. 2/3 if you switch is correct.

Dont you people read the proposition.

The proposition states that after the first door is opened there are only two doors left. The game show host will always open the door/s showing the goats. This problem is nothing to do with probabilitys and everything to do with ripping of the contestants.

1. Where is the "always" in the question".

2. It appears that you misunderstand what is going on. The (unstated but generally understood) rule of a game show is that you get whatever is behind the door of your choice at the time that door is opened. Whatever host does in between does not affect that. In fact, on the basis that, following your initial choice, he always opens a door to reveal a goat and offers you the opportunity to change your mind, he can only be helping you. If you change your mind, it improves your chances. If you stay put, your chances are unchanged.

You make a choice of one of three doors. The fact that the host opens the wrong door does not effect the out come of the game. You made your choice your stuck with it. Being able to change your choice at this point is incoherent logic. The opening of the doors in any order does not matter once the contestant makes his or hers choice. So why would you get to change up at this point It would totally effect the odds. It goes from a one in three to a 50/50 on the assumption that the choice you originally made is some how rendered non binding, but that's retarded. A 50/50 odds is better than a 1 in 3 but a 50/50 is exactly that, it's one or the other and what was there 5 min ago, is still there and changing your mind at this point is not going to effect that. The only thing it would succeed in is violating the game rules, so goat figure. I don't see the logic at any rate.

You made your choice your stuck with it.

On Let's Make a Deal, hosted by Monty Hall, you were not necessarily stuck with your first guess. It's a little like "Is that your final answer?" on Who Wants to be a Millionaire. Monty says something like: "Before I show you what's behind the door you picked, let me show you what's behind this door." He shows a goat. Then he says: "Do you want to stick with your original guess, or would you like to switch to the remaining door?"

More at: http://en.wikipedia.org/wiki/Monty_Hall_problem

Hmmm. Monty could vary the game - " I will reveal an unopened door and show you a goat , if you agree to a substitute star prize of lesser value than the original"

My proposed alternetive prize is a Gremlin , What should the original car prize have been ? <snigger> Two Goats ! hahahahahahahah

Your argument is tantamount to saying that additional information cannot be helpful. You would not make this argument if Monty showed you the correct door, closed it again and then gave you the option to change your mind. So the difference here is that the information (if there is any) is only partial.

If Monty's behaviour is consistent, and he always opens the door and this always reveals a goat then there is useful information. Other behaviours would have different consequences. As others of his ilk would say, "the choice is yours".

Forget about the odds! If the goats are good-looking they may be a better prize than the car. Signed, BigAl