A simple problem with a strange answer

Post Reply New Topic

So you use internal dialogue to solve this?

there is no chance that the hidden side can be the same as the visible side. it may look similar, but it is not the same side.

I initially and instinctively thought 1 in 3. But then I remembered there are two coins with the same side so I realised it must be 2 in 3.

OK, I didn't read anyone's replies before giving my answer as I didn't want to be influenced.

Now having read all the replies I'm so utterly confused. Maths is my weakest area and those convoluted replies made my head spin.

Did I get the right answer? I'm unsure but I think so.

If so, has anyone answered OPs question about why it's counter-intuitive? All I can say is that I originally thought it would be 1 in 3. So perhaps the thickest person maths-wise on the thread (moi) has answered this? E.g. because I intuitively thought 1 in 3 and then realised it was 2 in 3 so that was counter-intuitive.

Was my initial thought another way to perceive the problem? (What does it mean for ASD?)

I explained, or at least tried to, why I thought 2/3 was counter-intuitive, but I thought that 1/2 seemed the intuitive answer. How did you get 1/3?

i have found a solution to your question.
the answer is 1/3 !

i was going insane trying to work out an alternative answer to "2/3" because it was so obvious that that was the answer to your question as you worded it.

i tried to consider semantic aspects of your question, and found few avenues to produce an answer other than 2/3.

then i decided to google the question, and it was difficult because your question was worded by you, and was not copied from anywhere on the net, but i found after about 5 minutes, the question which should have been worded this way:

"A gambler is holding 3 coins. One is an ordinary quarter, the second has 2 tails, the third has 2 heads. The gambler chooses one of the coins at random and flips it, showing
heads. What is the likelihood that the other side is tails?"

you failed to state that the coined landed "heads" up, and so the correct answer is 1 in 3 and not 2 in 3. your rendition of the question allowed for an equal probability of either side landing face up because it would not matter with respect to your description of the problem.

whatever.

I didn't fail to state anything I wrote it exactly how it was written in the book and gave the title and author.

well i suppose that there may be variants on the example of that question that i found, but it is a simplification to remove the importance of the probability of it coming up either heads of tails in the consideration of the "problem"

the question as you worded it was so easily solvable that i had to question the obviousness of my automatic solution and try to find ways that i may be wrong. this is a hoax in my mind because i can not see how anyone would find the obvious (and correct) answer counter intuitive.

I did doubt my answer at first but not because of the question itself. The difficulty level stated as 7/10 and the title has the word "paradox" in it. The leads me to assume there is more to the question than meets the eye and that the answer is too simple.

I don't understand why the author says it's "counter-intuitive". It's interesting to see how people process this question, the in-depth discussion and the experiment. I think this is the only AS part to it though

I solve problems easiest when they are visual. After reading the problem I simplify it by removing the heads/tails aspect and rephrase it as: "there is 1 red coin and 2 blue coins. What's the probability of getting a blue coin?"

The only stumbling point to the question is that the heads and tails don't matter.

Respectfully, I believe the stumbling block was the element of mystery.

Most of the description is superfluous, if 2 of 3 coins will always be the same on both sides, no matter what their position, regardless of whether you are looking, and if you ignore the innuendo.

Because I thought there are 3 coins and you are going to be flipping 1 of them. So it's one coin out of three. I told you maths wasn't my strong point! I seem to have thought of an answer different to everyone else, hence with maths being my weak subject I presumed it wrong - which was more or less confirmed by everyone else seeming to say 2 in 3. Still, at least I rethought and came up with the right answer shortly after, so perhaps not such a lost cause (even though it's a very simple question!)

the stumbling block was the "illusion" of mystery for me.

I think a lot of you are misunderstanding the reasoning behind the solution. Revealing the topside of the coin in this case does not affect the final probability, but it does affect the way we get the answer. Imagine we have three coins, one a heads-tails coin, one a tails-tails coin and one a heads-heads coin (we'll call one side heads-a and the other heads-b).

Imagine now we pull one coin out of the hat and without looking place it on the table. We keep our hand over it so we cannot see the coin. At this point there are three possible coins, two of which are double sided. Therefore the coin has a 2/3 probability of being double sided. This is where I originally stopped and decided the answer was 2/3.

What originally missed is that the question asks us to remove our hand and look at the top side. Let's say in this case it was heads. From looking at the top side of the coin we can be certain that the coin is not the tails-tails coin, it has to be either the heads-heads coin or the heads-tails coin. There are two possible coins with only one being double-sided, thus the probability intuitively seems to be 1/2.



The incorrect, but intuitive, thinking is that when we put the coin on the table there were 3 possibilities:
1. The coin is heads-heads
2. The coin is heads-tails
3. The coin is tails-tails
By revealing the heads on the top side we can eliminate number 3 leaving the two possibilities, hence 1/2. 1/2 is the intuitive, but incorrect answer.



In actuality there were 6 possibilities when we placed the coin on the table:
1. The coin is heads-heads, with heads-a facing up
2. The coin is heads-heads, with heads-b facing up
3. The coin is heads-tails, with heads facing up
4. The coin is heads-tails, with tails facing up
5. The coin is tails-tails, with tails-a facing up
6. The coin is tails-tails, with tails-b facing up
This time by revealing heads we can eliminate numbers 4, 5 and 6, leaving three possibilities, two of which satisfy our condition. Hence 2/3 is the correct probability.

Seeing the top side does affect the probability, it's just a coincidence that in our problem we end up with the same answer. You could actually repeat the problem with one heads-heads coin, one heads-tails coin and 100 tails-tails coins, and if we revealed heads as the top side (however unlikely that would be) the probability would still revert to 2/3.

As for the 1/3 probability some people are mentioning, I don't think that comes into it at all. The problem b9 mentioned is different in that it is asking for the probability that the coin is a heads-tails coin, whereas our problem is asking for the probability of a double-sided coin. That's why the solution to b9's problem is 1/3 but the solution to ours is 2/3.

either way, it is a "no brainer" (as they say).
there is very little effort of thought required to solve the problem, and that fact engendered in me the suspicion that i was not seeing the full scope of the problem, and i wasted valuable time trying to dismantle what i thought was a very well encrypted puzzle, only to discover that my effort was in vain.

Very little thought required to come up with the correct answer, but I wouldn't necessarily call that solving the problem.

nothing is absolutely necessary. if you can not accomplish the construction of an ideational structure that you can present that verifies the logical validity of your position, then it is not "absolutely necessary" that you should "call" anything "anything.".