Monty Hall Problem... For Newbies

[Alferd_Packer]

If I understand correctly, Alferd_Packer is claiming that Monty's knowledge is an irrelevant detail that is included merely to make the problem more colorful. In essence his claim is that objecting "But Monty knows where the car is in the original Monty Hall problem!" is the same as objecting to someone who calls the host Morgan Henry. The host's name is completely irrelevant, and objecting to using the wrong name would be absolutely ridiculous, pathetic and stupid. Alferd thinks that Monty's knowledge in the classic problem is just like Monty's name, totally irrelevant. As has been shown, this is not true. Monty's knowledge actually makes a difference when it comes to the probabilities.

It's hard to admit that one is wrong, but on further reflection, I realize that I was indeed wrong.

If Monty's choice was random, I was ignoring those choices when he revealed a car and focusing solely on the choices when he reveled a goat.

Mea Culpa.

 

[JohnnyG]

I think JonnjyG’s simulation was based on the assumption that the question asked relates to the overall odds of a multiple toss. Whether this assumption is true or not is what’s being debated.

My simulation is a logical implementation of the original statement that "at least one of them landed heads." The statement is quite clear, I don't see how there is any assumption up for debate.

 

[ynot]

In the former the odds of 2 heads is 1/3, in the latter it is 1/2.

In the former you do not know which of the 2 coins landed heads. The important phrase there is "at least one"; it is not specifically talking about just one coin. That still leaves open the possibility of either of them being the coin to being known as having landed heads.

So what? The asker doesn't ask which of the two coins is heads. (s)he merely asks if one is heads. Given "it is known that at least one of the coins landed heads" it is established that the result of at least one coin toss has been viewed and is therefore known by the looker who then passes it on to the asker.

In the latter you do know which coin landed heads. The important phrase there is "one of the coins is known"; it is specifically talking about just 1 coin. At that point you are merely asking for the odds of the other coin, which of course is 50/50.

"one of the coins is known to have landed heads" does not establish to the asker which coin landed heads any more or less than the other scenario does. In both scenarios the asker don't know anything until the looker tells. In neither scenario the looker doesn’t tell the asker which coin was heads because that information wasn’t asked for. The looker doesn't answer the question by calculating the odds, (s)he merely looks at the empirical results and reports the result within the scope of the question (which doesn't include which coin).

 

[ynot]

Am I getting trolled again?

I don't know. Am I?

 

[Paul C. Anagnostopoulos]

Monty giving you the goat door is essentially the same as Monty offering you both doors. Realising that fact is a good way of highlighting the advantage of switching.

Right, but explaining it is tricky. What is wrong with jiggeryqua's objection:

"Largely unargued. But... Monty doesn't offer you both doors, and I can only speak from my conceptual framework, but claiming he's offering you both the goat and the (2/3 car)-door doesn't wash. He's also, at the same time and equally ('50/50') offering you your door plus the goat door. You can change to the other door and see a goat, or keep your door and see a goat. 50/50, right? Wrong - so, with the best will in the world, the explanation must be wrong. Right?"

The problem is that I don't want my door, I want the other two doors. So we have to make it clear that Monty needs to somehow offer me both of the other doors, not just any two doors.

~~ Paul

 

[ynot]

?

Again, I didn't say that. You have an extra "not" there.

Well is this better for you? - "Since when have we not been able to look at an individual coin without being able to know how it landed?"

 

[ynot]

My simulation is a logical implementation of the original statement that "at least one of them landed heads." The statement is quite clear, I don't see how there is any assumption up for debate.

But I do, and that's why I'm debating.

It's possible to know that "at least one of them landed heads." by looking at just one coin (but not always).

 

[HumptyDumpty]

I lot of discussion on the 2 coin problem I see, but 1st to reply to some comments made on the MHP.

Since the OP specified Monty knows where the car is, case B is irrelevant. And case C requires the use of information not in the OP, so it is also irrelevant.

Should we also explore the cases where the original car location is not random?

As I've already said, I included case B to show how looking at the reason why Monty opens the door he does leads to the answer of 50/50 (as Alferd has now realised). As to case C, the OP doesn't state how Monty chooses which goat door to open, and since different assumptions about his goat opening algorithm lead to different answers I thought it was relevant.

Even then your overall odds of winning the game will still be 2/3 if you play properly. But if you reassess your odds after seeing which door Monty opens (i.e. whether or not it is his preferred) then they will either be 1/2 or 1.

Correct. Overall you'll still win on average 2 times out 3 by always switching, but in any individual trial (in the Preferred Door variant) the probability the car is behind the door Monty didn't open is either 1/2 or 1, but is not 2/3 (that is only the case if Monty picks a goat door at random)

As for the 2 coin problem, the answer depends on what assumptions you make (just like the MHP ) I haven't read all the responses in detail but Brian-M has posted the most accurate analysis.

 

[JohnnyG]

There are two possible scenarios here, each with different odds. The description of what's going on is ambiguous, it could apply for either situation. The answers people give going to depend on which scenario they assume is being talked about.

Let's say X represents either heads or tails.

Scenario A:

You pick a value for X at random
You flip a pair of coins until at least one coin lands X
You then say "At least one coin is X"
The odds of both coins being X is 1/3​

Scenario B:

You flip a pair of coins.
You set X as the value of an arbitrarily selected coin
You then say "At least one coin is X"
The odds of both coins being X is 1/2​

In scenario B the setup does not match the logic. If the selected coin was not X, but the other coin was, then the statement "at least one coin is X" is true but is not said. The statement that "at least on coin is X" is the logical equivalent of "not both coins are not X". To accurately make that assertion requires looking at both coins.

 

[JohnnyG]

But I do, and that's why I'm debating.

It's possible to know that "at least one of them landed heads." by looking at just one coin (but not always).

The "but not always" part here is critical. If do not make that statement in all cases when it is true then it is not an accurate logical description of the situation.

I do agree that the problem can be worded to make it ambiguous, and when done that way it becomes a problem of interpretation, not one of probability. But this wording is not ambiguous.

 

[ynot]

In scenario B the setup does not match the logic. If the selected coin was not X, but the other coin was, then the statement "at least one coin is X" is true but is not said. The statement that "at least on coin is X" is the logical equivalent of "not both coins are not X". To accurately make that assertion requires looking at both coins.

The original scenario in this thread clearly defined that both coins had been examined . . .

'If I toss two fair coins, examine both of them and truthfully tell you that at least one of them landed heads. What is the probability both coins landed heads?'

 

[ynot]

The "but not always" part here is critical. If do not make that statement in all cases when it is true then it is not an accurate logical description of the situation.

I do agree that the problem can be worded to make it ambiguous, and when done that way it becomes a problem of interpretation, not one of probability. But this wording is not ambiguous.

But as we both agree we know that both coins have been examined.

 

[JohnnyG]

As I've already said, I included case B to show how looking at the reason why Monty opens the door he does leads to the answer of 50/50 (as Alferd has now realised). As to case C, the OP doesn't state how Monty chooses which goat door to open, and since different assumptions about his goat opening algorithm lead to different answers I thought it was relevant.

If nothing else, your scenarios do show that changing the problem by adding or ignoring known information can affect the outcomes.

 

[JohnnyG]

But as we both agree we know that both coins have been examined.

Yes, thank you for pointing that out. The statement that "at least one landed heads" implies examining both may be needed for completeness, but I had forgot that that the OP explicitly said it.

 

[ynot]

Yes, thank you for pointing that out. The statement that "at least one landed heads" implies examining both may be needed for completeness, but I had forgot that that the OP explicitly said it.

We are told the result of only one coin toss even though the results of both tosses are known. In my opinion that separates one toss from the other. If both coin tosses were being grouped together by the question it would have been something like - “Both coins landed heads, what’s the odds of the happening?”

Long threads like this can often suffer from a Chinese Whispers syndrome.

 

[Modified]

The original scenario in this thread clearly defined that both coins had been examined . . .

"... examine both of them and truthfully tell you that at least one of them landed heads" still doesn't tell you what game you're playing. If one is heads and one is tails, will you always say the same, or sometimes say "at least one of them is tails"? If there are two tails, will you say "both are tails / neither are heads" or "at least one of them is tails". You need to know the rules before you can determine the odds.

 

[ynot]

Think I'm pretty much done with both the coins and the Monty.

 

[ynot]

"... examine both of them and truthfully tell you that at least one of them landed heads" still doesn't tell you what game you're playing. If one is heads and one is tails, will you always say the same, or sometimes say "at least one of them is tails"? If there are two tails, will you say "both are tails / neither are heads" or "at least one of them is tails". You need to know the rules before you can determine the odds.

Why would anyone that wants the tosses to be grouped in the answer not give the grouped results in the scenario the question is related to?

 

[HumptyDumpty]

"... examine both of them and truthfully tell you that at least one of them landed heads" still doesn't tell you what game you're playing. If one is heads and one is tails, will you always say the same, or sometimes say "at least one of them is tails"? If there are two tails, will you say "both are tails / neither are heads" or "at least one of them is tails". You need to know the rules before you can determine the odds.

Very perspicacious. Just as, will Monty always open Door2 (if the car is behind Door1) or will he sometimes open Door3?

 

[Modified]

Why would anyone that wants the tosses to be grouped in the answer not give the grouped results in the scenario the question is related to?

I don't know what you're asking. Why don't you just state the full rules of the game, and I'll tell you the odds?