Three boxes and an Indian couple

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duszek
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Three boxes and an Indian couple

Post by duszek » Wed Aug 28, 2019 7:01 pm

In a novel by McEwan "The Sweet Tooth" a problem of logic and probability shows up:

There are three closed boxes, one contains a wonderful prize.

A contestant is asked to choose one but not open it yet. The show master (who knows where the prize is) opens one of the two remaining ones which is empty.
Now: shall the contestant stay with his first choice or shall he switch to the other unopened box ?

I can follow so far, yes the probability is higher when he moves to the other unopened box.

But what I cannot understand is this:
A writer makes a story using this problem.
A jealous husband goes to a hotel, there are three closed rooms, in one of them there is his wife and her lover. He chooses one closed room, then he hears a noise, hides, from one of the other two rooms an Indian couple goes out and leaves the hotel. Now the husband switches to the other closed room because this is the more probable one with the adulterous couple.

The heroine in the novel objects to this story because the Indian couple is not like the show master, they don´t know about which room contains the wife and her lover.

I don´t understand why it matters for the probability.
Can someone help ?

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vegetariantaxidermy
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Re: Three boxes and an Indian couple

Post by vegetariantaxidermy » Thu Aug 29, 2019 1:03 am

I've always thought that the 'Monty Hall problem' counldn't apply to the real world. Humans are not mathematical equations. Here's what Monty Hall had to say about it:

''Hall gave an explanation of the solution to that problem in an interview with The New York Times reporter John Tierney in 1991.[28] In the article, Hall pointed out that because he had control over the way the game progressed, playing on the psychology of the contestant, the theoretical solution did not apply to the show's actual gameplay. He said he was not surprised at the experts' insistence that the probability was 1 out of 2. "That's the same assumption contestants would make on the show after I showed them there was nothing behind one door," he said. "They'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered. By opening that door we were applying pressure. We called it the Henry James treatment. It was 'The Turn of the Screw.'" Hall clarified that as a game show host he was not required to follow the rules of the puzzle as Marilyn vos Savant often explains in her weekly column in Parade, and did not always have to allow a person the opportunity to switch. For example, he might open their door immediately if it was a losing door, might offer them money to not switch from a losing door to a winning door, or might only allow them the opportunity to switch if they had a winning door. "If the host is required to open a door all the time and offer you a switch, then you should take the switch," he said. "But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.''

Age
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Re: Three boxes and an Indian couple

Post by Age » Thu Aug 29, 2019 6:32 am

duszek wrote:
Wed Aug 28, 2019 7:01 pm
In a novel by McEwan "The Sweet Tooth" a problem of logic and probability shows up:

There are three closed boxes, one contains a wonderful prize.

A contestant is asked to choose one but not open it yet. The show master (who knows where the prize is) opens one of the two remaining ones which is empty.
Now: shall the contestant stay with his first choice or shall he switch to the other unopened box ?

I can follow so far, yes the probability is higher when he moves to the other unopened box.
Why is it said that the probability is higher when they move to the other unopened box?

duszek
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Re: Three boxes and an Indian couple

Post by duszek » Thu Aug 29, 2019 12:52 pm

Age wrote:
Thu Aug 29, 2019 6:32 am
duszek wrote:
Wed Aug 28, 2019 7:01 pm
In a novel by McEwan "The Sweet Tooth" a problem of logic and probability shows up:

There are three closed boxes, one contains a wonderful prize.

A contestant is asked to choose one but not open it yet. The show master (who knows where the prize is) opens one of the two remaining ones which is empty.
Now: shall the contestant stay with his first choice or shall he switch to the other unopened box ?

I can follow so far, yes the probability is higher when he moves to the other unopened box.
Why is it said that the probability is higher when they move to the other unopened box?

duszek
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Location: Thin Air

Re: Three boxes and an Indian couple

Post by duszek » Thu Aug 29, 2019 12:57 pm

Age wrote:
Thu Aug 29, 2019 6:32 am
duszek wrote:
Wed Aug 28, 2019 7:01 pm
In a novel by McEwan "The Sweet Tooth" a problem of logic and probability shows up:

There are three closed boxes, one contains a wonderful prize.

A contestant is asked to choose one but not open it yet. The show master (who knows where the prize is) opens one of the two remaining ones which is empty.
Now: shall the contestant stay with his first choice or shall he switch to the other unopened box ?

I can follow so far, yes the probability is higher when he moves to the other unopened box.
Why is it said that the probability is higher when they move to the other unopened box?
I had the same problem at first. And the writer in the novel too.

Let´s assume there are one million boxes.
You are allowed to pick one at first. Chance of 1 to a million to pick the one with the prize in it.

Another person picks one million minus 1 box, that is all the rest. He gets a huge amount of more chances of picking the right box.

Then the show master opens a million boxes minus two which are empty, on the other person´s side.

Now where is the prize ? In your box or in the box left unopen from the lot of the other person ?

:D :D

:D Take your time, don´t rush.

mickthinks
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Re: Three boxes and an Indian couple

Post by mickthinks » Thu Aug 29, 2019 2:24 pm

All you ever wanted to know about the Monty Hall Problem can be found here : https://en.wikipedia.org/wiki/Monty_Hall_problem , and the answer to Mc Ewan's problem can be deduced from there too.

You can also search this forum using "monty hall" and you'll find extensive discussion.

My quick and dirty attempt to answer duszek's question:

The heroine is not quite correct and therefore her objection is misleading. The case of the jealous husband is not the same as the Monty Hall problem, but it isn't so much because the Indian couple don't know where the adulterous couple are, but because there is no game for the Indian couple to take part in. To put it another way, when the Indian couple reveal themselves, that isn't in response to the cuckold's first choice of room. Note also, seeing their departure increases the cuckold's confidence that he chose the right room to 50:50, but these new better odds are not improved by him switching his choice after seeing them leave. It's 50:50 whichever of the two remaining rooms he chooses.

The problem of the Cuckold and the Indian couple is further problematised by it being a narrative in a novel (within another novel). Consider whether the novelist, or indeed McEwan, has decided which of the rooms the adulterous couple are in and has chosen to write the Indian couple into one of the other rooms and written his jealous husband choosing the third, ie. wrong one ... or the first, ie. right one. Of course then the thing ceases to be a mathematical question of chance altogether. It's only a Monty Hall type problem if it inhabits a probability space with a number of variations beyond the one that McEwan has given us.
Last edited by mickthinks on Thu Aug 29, 2019 4:24 pm, edited 2 times in total.

duszek
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Re: Three boxes and an Indian couple

Post by duszek » Thu Aug 29, 2019 2:42 pm

Thanks Mick Thinks, but I still don´t understand.

What´s the difference between an empty box opening itself and a room showing itself as not the right one because the wrong couple leave it by chance ?

If it´s a million rooms and couples leave the wrong rooms one by one ... is the chance at the end 50 to 50 ?

mickthinks
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Re: Three boxes and an Indian couple

Post by mickthinks » Thu Aug 29, 2019 4:19 pm

Empty boxes don't do that kind of thing as a rule. Is the room showing itself as the wrong one? I don't think so. The couple leave and that shows the room to be the wrong one.

Now you may think this is quibbling of the meaning of "showing". It isn't a qibble because it isn't a matter of meaning, it's a matter of mechanism or means, and that is of crucial importance in most probability models.

However, reviewing my analysis I realise it doesn't really answer your question. I'll think about it and try again.

In the meantime, yes, if it´s a million rooms and couples leave the wrong rooms one by one the chance at the end is 50 to 50. But the chance of that happening (in real life as opposed to in a novel) are millions, or is it billions or trillions, to one against (I think)

Age
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Re: Three boxes and an Indian couple

Post by Age » Tue Sep 03, 2019 5:38 am

duszek wrote:
Thu Aug 29, 2019 12:57 pm
Age wrote:
Thu Aug 29, 2019 6:32 am
duszek wrote:
Wed Aug 28, 2019 7:01 pm
In a novel by McEwan "The Sweet Tooth" a problem of logic and probability shows up:

There are three closed boxes, one contains a wonderful prize.

A contestant is asked to choose one but not open it yet. The show master (who knows where the prize is) opens one of the two remaining ones which is empty.
Now: shall the contestant stay with his first choice or shall he switch to the other unopened box ?

I can follow so far, yes the probability is higher when he moves to the other unopened box.
Why is it said that the probability is higher when they move to the other unopened box?
I had the same problem at first. And the writer in the novel too.

Let´s assume there are one million boxes.
You are allowed to pick one at first. Chance of 1 to a million to pick the one with the prize in it.

Another person picks one million minus 1 box, that is all the rest. He gets a huge amount of more chances of picking the right box.

Then the show master opens a million boxes minus two which are empty, on the other person´s side.

Now where is the prize ?
I do not know.
duszek wrote:
Thu Aug 29, 2019 12:57 pm
In your box or in the box left unopen from the lot of the other person ?

:D :D

:D Take your time, don´t rush.
I do not know, but I will ask the same question; Why IF they or I move to the other box it is said the probability is higher?

To me the probability stays the exact same, that is 50% chance. There are two boxes. To me, the prize could be in either one, so WHY does just changing boxes it is then said the probability is higher?

Of course the probability of a prize being in one of two boxes is higher than a prize being in one of a million boxes. This speaks for itself, but my question is in regards to what you previously wrote.

duszek
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Re: Three boxes and an Indian couple

Post by duszek » Wed Sep 04, 2019 5:36 pm

Mr Age

how about this:

two people are in front of 1 000 000 boxes.

Person A picks only one box, person B gets 999 999 remaining boxes.

In whose lot is a greater chance of being the prize ?

Now the show master (who knows which boxes are empty) opens 999 998 boxes of B´s lot.

Whose box has higher probability of containing the prize ?

Perhaps you look at the problem before the boxes get open. Only one for A (one chance in a million), 999 999 for B.
Or: A gets one pick, B gets 999 999 picks. Who is more likely to find the prize ?

The prize does not switch its position after empty boxes on B´s side get open.

duszek
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Re: Three boxes and an Indian couple

Post by duszek » Wed Sep 04, 2019 5:43 pm

Mick, I have an idea.

If wrong boxes or rooms open themselves then the room picked at first could also open itself and thus reduce the chances to 50 to 50 for the remaining two ones.

The show master opens an empty box leaving out the first choice.

That´s why the mathematician in the novel tried to improve the story this way:
The jealous husband hears two maids talking about cleaning the empty rooms on the floor in question. The husband stands at the room chosen by him (pretending that either he is a friend of the adulterous couple or he got the room and wants to check in himself).
The maid comes, passes "his" room and goes to one of the other two.

Now, can the maid replace the show master ? By a trick she was made to omit the room chosen by the husband and she goes to one of the empty ones or the only empty one.

Age
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Re: Three boxes and an Indian couple

Post by Age » Thu Sep 05, 2019 4:11 pm

duszek wrote:
Wed Sep 04, 2019 5:36 pm
Mr Age

how about this:

two people are in front of 1 000 000 boxes.

Person A picks only one box, person B gets 999 999 remaining boxes.

In whose lot is a greater chance of being the prize ?
Person B.
duszek wrote:
Wed Sep 04, 2019 5:36 pm
Now the show master (who knows which boxes are empty) opens 999 998 boxes of B´s lot.

Whose box has higher probability of containing the prize ?
Person B.
duszek wrote:
Wed Sep 04, 2019 5:36 pm
Perhaps you look at the problem before the boxes get open. Only one for A (one chance in a million), 999 999 for B.
Or: A gets one pick, B gets 999 999 picks. Who is more likely to find the prize ?
Person B.
duszek wrote:
Wed Sep 04, 2019 5:36 pm
The prize does not switch its position after empty boxes on B´s side get open.
Okay.

So, now what is the answer to the question that I asked you;
Why is it said that the probability is higher when they move to the other unopened box?


Is the answer some thing like; The probability is not higher 'when they move' to the other unopened box, but rather, the probability that the prize is in the other unopened box is higher only because originally there was more chances that the prize was in the 'other' unopened box, and only because this is a game and the one person who KNOWS where the prize is has now eliminated ALL of the non prize bearing boxes?

Is this some thing like the answer to my clarifying question?

Age
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Re: Three boxes and an Indian couple

Post by Age » Thu Sep 05, 2019 4:54 pm

duszek wrote:
Wed Aug 28, 2019 7:01 pm
In a novel by McEwan "The Sweet Tooth" a problem of logic and probability shows up:

There are three closed boxes, one contains a wonderful prize.

A contestant is asked to choose one but not open it yet. The show master (who knows where the prize is) opens one of the two remaining ones which is empty.
Now: shall the contestant stay with his first choice or shall he switch to the other unopened box ?

I can follow so far, yes the probability is higher when he moves to the other unopened box.

But what I cannot understand is this:
A writer makes a story using this problem.
A jealous husband goes to a hotel, there are three closed rooms, in one of them there is his wife and her lover. He chooses one closed room, then he hears a noise, hides, from one of the other two rooms an Indian couple goes out and leaves the hotel. Now the husband switches to the other closed room because this is the more probable one with the adulterous couple.

The heroine in the novel objects to this story because the Indian couple is not like the show master, they don´t know about which room contains the wife and her lover.

I don´t understand why it matters for the probability.
Can someone help ?
Now, if my last response is somewhat correct, then the reason why it matters for the probability is because there is no one knowing.

In this case going to the 'other' closed room is NOT more probable. The couple that just left have NO idea of what is going on. Whereas a host of a 'game' who KNOWS what is inside is a completely different scenario.

Age
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Re: Three boxes and an Indian couple

Post by Age » Thu Sep 05, 2019 5:01 pm

duszek wrote:
Wed Sep 04, 2019 5:43 pm
Mick, I have an idea.

If wrong boxes or rooms open themselves then the room picked at first could also open itself and thus reduce the chances to 50 to 50 for the remaining two ones.

The show master opens an empty box leaving out the first choice.

That´s why the mathematician in the novel tried to improve the story this way:
The jealous husband hears two maids talking about cleaning the empty rooms on the floor in question. The husband stands at the room chosen by him (pretending that either he is a friend of the adulterous couple or he got the room and wants to check in himself).
The maid comes, passes "his" room and goes to one of the other two.

Now, can the maid replace the show master ? By a trick she was made to omit the room chosen by the husband and she goes to one of the empty ones or the only empty one.
In this scenario the maid KNOWS where the empty room is or rooms are. But the probability still remains the same and is NOT higher like in the game show scenario. When the maid goes into one room there is NO reason for the husband to change rooms because the probability of his wife being in the other room is NOT higher. The maid MIGHT KNOW if the other room is empty or not. But, the husband does NOT know if the maid KNOWS. Whereas, the contestant KNOWS that the game show host does KNOW which box contains the prize.

duszek
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Re: Three boxes and an Indian couple

Post by duszek » Mon Sep 09, 2019 4:48 pm

Mr Age,
thank you for your attention to this problem that I am still struggling with myself.

The maid was invented to make the original choice or pick left aside.
If it was simply for one of the wrong rooms disclosing themselves as wrong then the Indian couple might have left from the original pick of the husband.

Why switching to another box or room ?
Because of the calculation of probability.

There is only one prize in one of three boxes, would you pick one box or two boxes hoping for the right pick ?
Even without a show master opening an empty box of the two together it is better to switch to two boxes instead of staying with one.
So yes, your answer to your underlined question is what I also think is correct way of looking at it.

But I still don´t quite see why we are not allowed to calculate the probability anew each time one empty box is being removed from the pool of options.
I am still not quite converted myself, if I am honest.

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