A Fistful of Brain Teasers

Innocent Bystander posted:

Don Atkinson posted:

!

 

Monty Hall stopped the show once they realised that contestants realised that to "swap" was the best strategy.

 

But that seems to confirm that the show was rigged, so nothing to do with chance, but to do with how the show was run - for which awareness of the show’s history, or at least of its designers’ psychology, is what makes the switch choice work - not that it is 2/3

The show wasn't rigged.

Eloise posted:

Innocent Bystander posted:

But that seems to confirm that the show was rigged, so nothing to do with chance, but to do with how the show was run - for which awareness of the show’s history, or at least of its designers’ psychology, is what makes the switch choice work - not that it is 2/3

Of course game shows are rigged ...

There is always an exception !

I never did like game shows. Slumdog Millionaire anyone?

Don Atkinson posted:

Innocent Bystander posted:

But that seems to confirm that the show was rigged, so nothing to do with chance, but to do with how the show was run - for which awareness of the show’s history, or at least of its designers’ psychology, is what makes the switch choice work - not that it is 2/3

The show wasn't rigged.

Well it was in a way ... basically it was better for the show and the audience if the contestant had a better chance of winning the car - even if they didn't know.

But to explain further ... 

Doors 1, 2 and 3 - so a 1/3 chance of winning.  The contestant can choose any door initially so lets say door 3 is chosen.  The possible options are

1. Car - Goat - Goat
2. Goat - Car - Goat
3. Goat - Goat - Car

The host (Monty) will always then open the door which is both un-chosen and doesn't have the car behind it (its a flop for ratings if its revealed too early the contestant hasn't won. 

The thing is though ... each door STILL has a 1/3 chance of being where the Car is; however there was a 2/3 chance of the car being behind door No.1 or No.2; and Monty would open the one where the car wasn't.  So the chance of the car being behind door No.1 or No.2 are combined into the remaining door (No.2).

You don't think of it as a new choice (which would be 50-50); you have to think of it as continuation of your first choice.  Of course swapping might be the wrong thing in any given situation, but mathematically its correct.

Eloise posted:

Don Atkinson posted:

Innocent Bystander posted:

But that seems to confirm that the show was rigged, so nothing to do with chance, but to do with how the show was run - for which awareness of the show’s history, or at least of its designers’ psychology, is what makes the switch choice work - not that it is 2/3

The show wasn't rigged.

Well it was in a way ... basically it was better for the show and the audience if the contestant had a better chance of winning the car - even if they didn't know.

But to explain further ... 

Doors 1, 2 and 3 - so a 1/3 chance of winning.  The contestant can choose any door initially so lets say door 3 is chosen.  The possible options are

1. Car - Goat - Goat
2. Goat - Car - Goat
3. Goat - Goat - Car

The host (Monty) will always then open the door which is both un-chosen and doesn't have the car behind it (its a flop for ratings if its revealed too early the contestant hasn't won. 

The thing is though ... each door STILL has a 1/3 chance of being where the Car is; however there was a 2/3 chance of the car being behind door No.1 or No.2; and Monty would open the one where the car wasn't.  So the chance of the car being behind door No.1 or No.2 are combined into the remaining door (No.2).

You don't think of it as a new choice (which would be 50-50); you have to think of it as continuation of your first choice.  Of course swapping might be the wrong thing in any given situation, but mathematically its correct.

How? It is known not to be behind 1, so is either behind 2 or 3. At that point your original choice is irrelevant because in effect you are starting again, just choosing between 2 options, so the chance is 1/2. How is that flawed?

Incidentally, did the presenter always give that option, regardless of whether the contestant had chosen the right one? And was there any difference in his manner/body language in the different scenarios?

I don't think that it is as simple as a case of pure statistical probabilty.

As Don says the choice of the first door opened is biased based on 2 factors i.e. the presenter will not open the door selected by the contestant (whether he is correct or not) neither will he open the door which has the car behind it.  If the contestant's first guess was correct (an initial 1/3 probabilty) then the presenter can open either of the other 2 doors. However, if his initial guess was wrong (a 2/3 probability) then the presenter has only one option which is to open the door which is neither the selected door nor the door with the car behind it. 

On the basis that probability suggests that the contestant's first guess will be wrong (1/3 as against 2/3) then logically he should switch.

P.S. it seems most unlikely that the presenter would not know which door held the car behind it.

Hi IB,

you may want to consider one or both of these thoughts:

Even with one door open, you can still choose from all three. If you told your buddies at the Pub the night before that you would rather have a goat than one of those silly Porsches, pick door one and be happy (or kick yourself for having one pint too many).

Your choice of door 1 determines the hosts action. Assuming that he will not show you the car, for the reason stated above, he has no choice if you picked one of the goats. He has to show you the other one.  Only if you have picked the car, he is free to pick any of the other two doors.

Mulberry posted:

Hi IB,

you may want to consider one or both of these thoughts:

Even with one door open, you can still choose from all three. If you told your buddies at the Pub the night before that you would rather have a goat than one of those silly Porsches, pick door one and be happy (or kick yourself for having one pint too many).

Your choice of door 1 determines the hosts action. Assuming that he will not show you the car, for the reason stated above, he has no choice if you picked one of the goats. He has to show you the other one.  Only if you have picked the car, he is free to pick any of the other two doors.

So, the argument appears to be: if I picked the goat the host first has to open the empty one - and change I must to win. If I picked the empty one, the host first has to open the one with the goat, and again change I must to win. If I picked the car, he can choose either, when I must not change to win. That is two scenarios when I mustn’t change and one where I must, so a 2/3 chance at the swap stage after the first door is opened, therefore better to change.

However, there is a flaw in this argument, in that the third scenario above is actually 2 scenarios rolled into one, because in practice If I picked the car, he can choose to reveal the goat, when I must not change to win. And If I picked the car, he can equally choose to reveal empty one, when I must not change to win. That makes 2 scenarios when I must change, and two when I mustn’t, giving an equal chance of picking the right one at the swap stage.

Innocent Bystander posted:

Eloise posted:

Don Atkinson posted:

Innocent Bystander posted:

But that seems to confirm that the show was rigged, so nothing to do with chance, but to do with how the show was run - for which awareness of the show’s history, or at least of its designers’ psychology, is what makes the switch choice work - not that it is 2/3

The show wasn't rigged.

Well it was in a way ... basically it was better for the show and the audience if the contestant had a better chance of winning the car - even if they didn't know.

But to explain further ... 

Doors 1, 2 and 3 - so a 1/3 chance of winning.  The contestant can choose any door initially so lets say door 3 is chosen.  The possible options are

1. Car - Goat - Goat
2. Goat - Car - Goat
3. Goat - Goat - Car

The host (Monty) will always then open the door which is both un-chosen and doesn't have the car behind it (its a flop for ratings if its revealed too early the contestant hasn't won. 

The thing is though ... each door STILL has a 1/3 chance of being where the Car is; however there was a 2/3 chance of the car being behind door No.1 or No.2; and Monty would open the one where the car wasn't.  So the chance of the car being behind door No.1 or No.2 are combined into the remaining door (No.2).

You don't think of it as a new choice (which would be 50-50); you have to think of it as continuation of your first choice.  Of course swapping might be the wrong thing in any given situation, but mathematically its correct.

How? It is known not to be behind 1, so is either behind 2 or 3. At that point your original choice is irrelevant because in effect you are starting again, just choosing between 2 options, so the chance is 1/2. How is that flawed?

Incidentally, did the presenter always give that option, regardless of whether the contestant had chosen the right one? And was there any difference in his manner/body language in the different scenarios?

 

 

Re your 1st paragraph - The contestant doesn't know where the car is. His initial choice of door is definitely 1/3. I think you accept that ? I will leave it at that for the moment, but it is important, although in a somewhat negative way.

Re your second paragraph - the presenter

          a) always opened a door to reveal a goat

          b) always gave the contestant the option to "change his mind"

I have put "change his mind" in inverted commas because this aspect is also very, very important. And using those words without quotes would be a bit misleading.

Knowing what you now know, try to develop a strategy that will give you a 2/3 chance of selecting the right door, right from the start..................

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

FWIW, I'm pretty sure that Monty Hall and the "Let's Make a Deal" show were long retired when this puzzle became a thing. Tapping into my childhood memories, the opportunity to change your mind was NOT a part of the show.

Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

Innocent Bystander posted:

Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

That was a fairly common comment made by directors.

Innocent Bystander posted:

winkyincanada posted:

Don Atkinson posted:

Alright, here's an old one so I hope you haven't heard the answer already. If you haven't it is quite interesting. Enjoy!

Late one cold, rainy night in November after a few pints and a rather dubious curry you flop down onto your favourite listening couch and turn on the TV. Ah...that's better. But Have I Got News For You isn't enough to keep your eyes from closing and you soon fall into a deep, deep slumber and start to dream..

Congratulations! You have been selected as a participant in a very popular national TV gameshow. This is your first time on TV and on such a show and you are overwhelmed with excitement and nervousness.

You find the first rounds easy and before long the other contestants have been eliminated and you have made it to the final challenge. The audience go wild! What a night!

Now the final challenge begins. The lights dim and the cheers of the crowd die down. Ant & Dec show you three doors numbered 1, 2 and 3. They say that the Porsche is hidden behind one of the doors. You must choose which one. If you choose correctly you win the Porsche. If not, you win a goat. The tension mounts. The audience can't contain themselves: shouts begin..."door 1"..."no door 2"...etc. You hesitate. Ant urges you to make a choice. Finally you choose door number 3. A green light comes on over door 3. Oh no...did I choose wisely...

A hush comes over the theatre as Dec walks over to door number 1 and motions to open it. The atmosphere can be cut with a knife. Slowly, Dec opens door 1 to reveal a goat! Phew that was lucky. The crowd bursts into a relieved cheer. But soon Ant has gripped the handle of door 2 and the crowd go silent. You can hear a pin drop.

The tension is unbearable...

Suddenly Ant stops turning the handle of door 2 and turns towards you. He says he is in a good mood tonight and wants to help you as much as he can. He offers to do you a favour and allow you to change your selection. You can now stick with your original choice...door 3 or switch to door 2. It is up to you and you have just 10 seconds to decide. The audience erupts! "door 2, go for door 2"..."no, no stick with door 3"...

You have 5 seconds left. You are sweating. Everyone is hanging on your next words. Do you stick with door 3 or switch to door 2?

As I said, it's an old one, so you no doubt know the best thing to do and that's fine - just post your option. But..

...top marks go to the best explanation for your decision to open Door No............

Always switch. You have 2/3 chance of winning if you switch. If you stick with your first choice, your chance of winning remains at 1/3, the same as it was when you selected it. Think of it this way. After you have chosen the first time, your door has a 1/3 chance of winning. The other two doors, combined, have a 2/3 chance of containing the car. Removing one of those doors  effectively "compresses" that 2/3 probability into the other door. So switch!

(This game (The Monty Hall game) depends on the host knowing which door contains the car, and always purposefully choosing a goat-door. Obviously, if the host opens a door at random, they will sometimes reveal the car. It changes the premise of the problem a bit, but whenever the host-opened door reveals a goat, you should switch.)

However, when you make that second choice the odds at that point are 1/2 not 1/3 because one has been eliminated, so it is just a toss up: why change? So the question is simply, will you have the courage of your convictions, or swap? However if you don’t switch and don’t win you will kick yourself all the more for having chosen wrong twice, so maybe that is a reason for change, if a bad one - it is an evil thing to offer you just for greater entertainment. But then, that is what game shows are about!

if it was me I’d think the presenter is trying to save the cost of the Porsche (ok, the motorised goat) and tempting me to think that I should have chosen door 2.

No. The probabilities are now 1/3 - 2/3, not 50:50. The likelihood that you were right the first time doesn't change. If there were a thousand doors, one car, and 999 goats, and the host opened 998 of them showing goats, would the chance that you picked the correct door the first time now be 50%? No, it stays at 0.1% and the likelihood that the car is behind the other door is 99.9%.

The answer is always to switch.

I don’t pretend that my explanation is any better than the explanation given by others but I set it out below nonetheless. I have already alluded to some aspects of it anyway.

Doors are labelled A, B and C

A car lies behind one door and a goat behind each of the other two

Given the choice of zero doors, the probability of getting the car is 0

Given a choice of only one door, the probability of getting the car is 1/3

Given a choice of all three doors, the probability of getting the car is 1 (it’s a dead cert !)

Given the choice of any two doors, the probability of getting the car is 2/3

That last option is crucial and………..it’s yours for the taking !

Instead of choosing C and sticking with it (1 in 3 chance)…..

…………..choose A and B

Yes, you heard right…..choose A and B

All you need now is to persuade the Show-Host to open both doors A and B

….and you then stand a 2/3 chance of winning that car !

And what’s so good about this is, you already know how to do it !

Youtell the Show-Host that you have “Chosen” Door C.

He then voluntarily opens either A or B (let’s say A)

You then tell him to open the other one ie B

No time to think about it at work, but I sussed it for myself cycling home (see, cycling is good for the brain, too!)

actually a very simple explanation:

At the time of picking, there is a 1/3 chance of it being behind the door I pick, so 2/3 chance that it will be behind  one or other of the other two doors? Rule out one of those two, and the 2/3 chance is now resting on just the one door, that I didn’t pick, my priginal choice still being 1/3.

THe evidence is now there, so I change my mind and swap!

Innocent Bystander posted:

No time to think about it at work, but I sussed it for myself cycling home(see, cycling is good for the brain, too!)

actually a very simple explanation:

At the time of picking, there is a 1/3 chance of it being behind the door I pick, so 2/3 chance that it will be behind  one or other of the other two doors? Rule out one of those two, and the 2/3 chance is now resting on just the one door, that I didn’t pick, my priginal choice still being 1/3.

THe evidence is now there, so I change my mind and swap!

I'm not yet convinced of that.

For sure it's true for the sort of off-road recreational cycling that I do (see my photos of Canada).

However, when people commute-cycle on our current road system, either knowing the risk involved or totally ignorant of the risk, I am inclined to think the opposite is true.

But that's for another thread

As it happens I was late today and there was virtually no traffic, and second half of journey none at all, hence freedom th think!

Don Atkinson posted:

Innocent Bystander posted:

Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

That was a fairly common comment made by directors.

And why should anyone reverse their decision without a reason to believe it is better than the original decisiom? You saying it is better doesnt make it so: what is needed is a weighing up of allavailable evidence, balance of probability of the new information being correct, and assessment of pros and cons of change. Given the facts as I reasoned in my explanation it is clear cut, but someone else saying without doing anything to show why their proposal is better is not a reason for making the change, though it can be a reason to consider it and make additional investigations or considerations.

Innocent Bystander posted:

Don Atkinson posted:

Innocent Bystander posted:

Don Atkinson posted:

I used to use this Monty Hall puzzle in some director training sessions.

I used a Red Ball and two Blue balls, placing them unseen to the directors, under three tins. (I knew where the Red ball was).

It was astonishing how many directors were psychologically unable to "change their mind" having selected "Tin C" (or whatever). Even after a couple of dozen runs, there were a few who couldn't cope. "I'm a DIRECTOR, once my mind is made up, I won't change" sort of mentality. Even when facts were laid bare in front of them.

The exercise was designed to encourage them to engage Experts to advise on matters they didn't understand themselves.

If the evidence suggests that the original decision was wrong, one should change it given the opportunity. Otherwise not (in the management example) or only if you are indecisive, or perceive some bias affecting things in the case of the game show. 

In this case there is no evidence that shows the original pick to be wrong. The best there seems to be is a conviction that the odds are 2/3 that car is behind the door that hasn’t been chosen or opened, so 1/3 that the original choice was correct. Unfortunately at this point it appears to me to be an unsubstantiated conviction by so eg people - which is not evidence...

 

That was a fairly common comment made by directors.

And why should anyone reverse their decision without a reason to believe it is better than the original decisiom? You saying it is better doesnt make it so: what is needed is a weighing up of allavailable evidence, balance of probability of the new information being correct, and assessment of pros and cons of change. Given the facts as I reasoned in my explanation it is clear cut, but someone else saying without doing anything to show why their proposal is better is not a reason for making the change, though it can be a reason to consider it and make additional investigations or considerations

I used the Car/Goat puzzle to encourage directors to seek the advice of experts. (see one of my earlier posts)

This puzzle demonstrates that a "gut" feeling can lead to inappropriate decisions. The concept is difficult to grasp. Don't make decisions until you understand the issues and likely outcomes. Accept the evidence of your experts. If you're not sure, ask two or three of them.

When I and others (in other words two or three experts) said it was better to switch, you knew that we had justifiable evidence. We simply hadn't yet revealed it. Even when revealed, many directors don't understand the evidence in front of them and have to trust their experts.

JRHardee posted:

FWIW, I'm pretty sure that Monty Hall and the "Let's Make a Deal" show were long retired when this puzzle became a thing. Tapping into my childhood memories, the opportunity to change your mind was NOT a part of the show.

I never saw this Monty Hall show and I only became aware of this "puzzle" some 15 years or so ago.

It's more than likely that a bit of "folk law" has developed around the show and the puzzle such that have become synonymous (is that the right word ?)

I hope we all enjoyed this one, I think it's a real Teaser.

Does anybody have a better idea than Mulberry's (brilliant) interpretation of NFG's teaser ?

My proposed solution is 47.53295m to a few more decimal places than before.

But I could be barking up the wrong tree.

Don Atkinson posted:

?)

I hope we all enjoyed this one, I think it's a real Teaser.

Indeed. 

Incidentally, I wonder how many of those who answered correctly had come across it before and learntbthe correct answer - or am I the only Dumbo who couldn’t see it? If so, no matter - it is always good to learn.

This will be especially useful if I ever find myself having to choose which path to cycle down, knowing one is nice and smooth and has no motorists or pedestrians, one joins a narrow road with speeding cars and pedestrians with dogs, and the third is a road with a cycle lane, but cyclists are taxed heavily for using it - when just after I have chosen one of them and am about to set off someone erects a sign on one of the others saying “Cycle route: toll £5 a mile”.

Arrows !!

Aaron, Bertie and Chuck have created their own Target Board.

Seven circles on a beige background. Each circle has its own points value as does the beige background which is so large, it simply can’t be missed !

Each player gets to fire six arrows at the Target Board.

Aaron made 25 with his first three shots. It was a coincidence that the third shot scored 3.

Chuck hadn’t scored at all, when Bertie picked up 2 points.

At the end of the tournament, 50 had been scored only once, 20, 10 and 1 were scored three times, and 25, 5, 3 and 2 were each scored twice.

To their utter amazement, despite all hitting different combinations of circles and background, all three participants wound up with the same score !

Who shot the Bull’s-Eye 50 ?

Bertie