A simple problem with a strange answer

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'So, if each sailor only paid $9 (to make $27), and the bellboy kept $2 (to make $29) what happened to the other dollar?'
^^This sentence was thrown in to throw you off. It's just a ruse. It's nonesense. There is no reason to add the $27 to the $2. There is no logic to it.

I have been thinking about this, and I think I have come up with an explanation for why this problem is identical to the Monty Hall problem.

Say you are the contestant and for the sake of argument you pick door number 3.

Lets call the two other doors 1) -> HEADS 2) -> TAILS. Monty Hall opens door 2 (TAILS) and shows you the goat.

There are two possible scenarios
a) a goat and a car, in which case Monty Hall MUST open the tails door. This is like giving Monty Hall a TAILS-TAILS coin to toss.

b) Two goats. In which case Monty Hall can open either door. This is like giving Monty Hall a (HEADS-TAILS) coin to toss.

Since a and b occur with equal likelihood, this is like saying that a coin has been tossed but you do not know whether it is the TAILS-TAILS or the HEAD-TAILS coin. All you know is that the result was TAILS.

Intuition would tell you that it is 50-50 but it is fact not. It is more likely that the coin that was tossed was TAILS-TAILS as other posters have already explained.

Therefore situation a) is more likely -> you should switch in order to get the car.

Is 2/3, this remind me to the monty hall problem.

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

I really enjoyed reading all of your posts, they've been an eye opener in more ways than I intended. As for this problem I think I understand the underlying reason why it's counter-intuitive. Because many people are unable to remove their sense of self and perception when solving this problem. Probability exists without human perception and does not change because of it. Even if a person can see all 6 faces of the coins the probability remains the same.

Another example that TO ME seems like a fit. If a tree falls in the forest does it make a sound? Of course it does because the mechanics of sound existed before people attached a word to it and will remain after people are gone. Like with the coins people have a hard time with the tree question because they have a difficult time removing 'self'. The difference is do you think of sound as a process or a perception, humans think of it in terms of perception.

Just a thought. Does this seem valid to you guys? Some have said this already but worded differently.

BTW if you like these sorts of 'puzzles' the book I mentioned in the OP is really good, I highly recommend it.

I've always found the tree question, and similar questions (one hand clapping) ridiculous. It always bothered me that people thought these sort of questions were confusing/amazing/thought provoking.

Sound is sound waves moving through stuff (air, gas, etc) in this case caused by the fall. Unless the forest was magically in a vacuum or something than it would make a sound. Nobody would hear it of course but it would still make it. Questioning if it made a sound is like questioning if said tree exists.

Now questioning if stuff exists when it's not observed is much more interesting and complicated question, but I'll leave that to the quantum physicists.

I almost added in my previous post "please do not start debating quantum mechanics"

Except that the other side isn't heads or tails, it's heads, heads or tails, so it's 2/3.

Coconuts hit on the right answer for the wrong reason.

If you pull out a coin and slap it on the table and see that it is (say) heads. Then you know that it cant be the coin with two tails. So you know it cant be one of the three coins. So that only leaves two possibilities (as I said on the first page of this thread). Its either the coin with two heads, or the one with both a head a tail. That means the other side of the coin can has a fifty fifty chance of being the same or being different than the visible side.

Problem one: its the same thing I said- you SEE one side of the coin after you pull it out, and that enables you to eliminate one of the three coins as possibility. Which in turn reduces the odds from 2/3 to fifty-fifty.

Problem two: someone said that what I was wrong and that you dont even have see the exposed side of the coin you pulled-thus you dont have to illiminate one of the duplicate sided coins. The odds are still fifty-fify - whether you know what the upside of the coin you pulled out is or not.

You are talking nonsense.

A coin that has A head cannot be a coin with two two tails.
So if you pull out a coin with a head- you can indeed eliminate the two tailed coin as a possibility. And that does indeed reduce the odds.

How can a coin with one head magically morph into a coin with two tails?

So..follow through to your conclusion PLEASE.

What, then, are the ODDS?

You have three possibilities- are you saying all three are equally likely?

If so- then your conclusion is 2/3 its going to be the same as whats visible. And 1/3 that it is different.

Is that what you are saying?

Let me explain what he is saying in different words. The question itself is very specific and involves three coins. The very fact that someone uses their own experience and knowledge to answer prevents them from answering the question. You see, if you eliminate one of the coins you are no longer answering a probability question about 3 coins. You are now answering a completely different question that involves two coins. This goes back to what I was saying earlier about the 'self' interfering with just seeing the raw facts.

Now I understand why the author wrote that many people find 2/3 counter-intuitive and how people find it counter-intuitive.

But the side you are looking at can either be one side of the double-heads, other other side of the double-heads, or the heads side of the heads-tails. So there's three possibilities.


Sorry! I actually left it there deliberately so people were free to draw their own conclusions from that, but yes, I see the odds as 2/3, because there are three possibilities and two of them are that they are the same on both sides.

It's an interesting problem. I also got the right answer for the wrong reason, because I understood the question to mean that you don't look at the coin at all, even after it's on the table.

I think another way to explain the logical flaw in the "1/2" solution is this: once you see heads, yes, it's true that there are 2 possibilities for which coin is on the table (heads/heads or heads/tails), but the incorrect assumption is that these possibilities are equally likely. They're not - it's twice as likely to be the heads/heads coin.

Ellingtonia explains it well, but this is a particularly important point that I think a few people still seem to be confused about.

Obviously "probability" here means "probability based on the information available to you". Even leaving quantum mechanics aside, there is no probability without human perception in this case. Once the coin is on the table it's bottom side is either 100% heads or 100% tails - it's just that you don't know which.

Okay.
2/3 IS indeed the right anwer.

Since 2/3 was the original answer I came up with in first post in this thread then I was right - right about the answer the puzzler.

Okay.

to anawer the OP's question- "Is the anwer 2/3 'counterintuitive"?- my answer to THAT question is NO. It is NOT counterintuitive.

For me it is NOT counterintuitive because its the first thing I thought of.

It was only when I worried about it being "counterintuitive" did go on to consider the possibility that the anwer to the puzzler might be 1/2 (instead of quitting while I was ahead).

So for me the right answer is my "intutive" answer.

Most people on this thread also posted 2/3 as the answer. Though a few did say that their first impulse was to say 50-50.

So... I dunno...since I (and most folks on this forum) are on the spectrum maybe it makes a difference.

Would nts go for 50-50 and aspies for 2/3? I dont know.

Ok-then you're wrong. In my opinion.

I go by the whole coin, not by the sides.

The reason is that we have already eliminated one whole coin (as a coin) as a possibility-namely the two tailed coin.

So the coin you have on the table is either the two headed coin, or its the normal coin with the head side up.

So what are the odds as to its identity?

Fifty-fifty. Because you're just as likely to have picked either coin at random.

So the odds are fifty-fifty that when you flip it over it will have a tail ( be different)or have another head (be the same).

Thus the odds are not 2/3, but 50-50.

Your reasoning would only work under the following circumstance:if you could physically detach the two sides of the coins from each other. That way the heads and tails would float around seperately, and have seperate destinies.

If the coin sides were not physically wedded to each other -then- at the first stage of the puzzler- you have the loose sides of all three coins floating around inside the hat. Since one coin has two tails, one has two heads, and one has both,- there would be six loose tails and six loose heads in the hat. So if you pulled out an amputated head of a coin- then the odds that you would match it with a second amputated head would be 2/5.

But you and I are not talking about that first stage of the puzzler, but the second stage- what happens AFTER you pull out the first coin and place it heads up on the table.


You arbitrarily just remove two tails out of the hat and toss em aside, and then place one loose head on the table.

This would be the equivalent of pulling out one intact coin and placing it on the table with one head up ( which eliminates the possibility that you have the two tailed coin on the table).

So now you have to determine the odds of matching that loose head with another loose head in the hat ( which is equivalent- but not the same- as flipping the intact coin with the head up).

Since weve already discarded two of the loose tails, and your visible head takes up one of the heads-that leaves: one remaining tail,and two remaining heads (like what you said).

So if you REACH BACK into the bag to grab one more hidden amputated side from a coin the odds that you match your head with another head is 2/3 since there are two remaining heads and one remaining tail- and the odds of randomly grabbing each is the same.

It would be the number you came up with.

But that is because each SIDE of the coin has been sawed off of the other side of the coin. Since the sides are physically detached- they have seperate destinies from their former mates. In the original puzzler the coins are not mutilated like that- so you have to track their destines as coins- and not as seperate sides.

Or atleast thats MHO.