A simple problem with a strange answer

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Oh, and:

[Moved from General Autism to Off the Wall.]

Maybe all we've been doing is Rascal's homework.

I'm dying to know the answer. Well, not dying, but...

It's not 50% because there is a greater chance that you selected a double sided coin. It has to be 2/3

Yes.

But the interesting question is this: Why does knowing what is on the top of the coin make the odds 50-50 while not knowing 2/3?

Think about it.

The coin is on the table but you haven't seen it yet. The odds are 2/3 that both sides will be the same.

Then you look at the top and the odds are 50-50 when nothing else has changed.

In other words, just having greater knowledge about the coin appears to change the odds dramatically.

Compare this to the Monty Hall problem. For those who might not be familiar with it, Monty Hall shows you three curtains on his game show and tells you that behind two curtains are goats and behind the third is something quite valuable, for example, a new car. You then pick a curtain. He then shows you what is behind another curtain and you have the option of changing curtains.

So you basically have a 1/3 chance of there being a car behind your curtain. From a naive viewpoint, when he shows you a curtain with a goat behind it, it appears to change the odds to 50-50. In reality, the odds haven't changed at all. You still have a 1/3 chance of there being a car behind your curtain and since you know there is a goat behind the curtain he just showed you, there is a 2/3 chance that the car is behind the other curtain that you did not choose.

So why does it appear to change the odds in this case when you look at the top of the coin from 2/3 to 50-50?

I think that the answer is because the question changed but you didn't notice. When you see the top of the coin, for the purpose of the discussion assume it is a heads, then question suddenly becomes not one of what are the odds that the other side is the same as the first but what are the odds that the other side is a heads. And that chance is 50-50.

If you go back to the original question and ask not whether or not the two sides are the same but what is the odds that the bottom side is a heads, you will find out that it was 50-50.

When you see the top of the coin, the question itself changes.

Knowing what is on top changes N-O-T-H-I-N-G.

Seriously, think about it. The question has nothing to do with what you know or do not know. The question is very simple.

"what are the chances that the hidden side is the same as the visible side?"

Where does what you see, or what you know play into the answer? Neither does.

The factors of placing the coin on the table and of not looking at it, are called red herrings. <<< link

This reminds me of the Monty Hall Problem although I seem to recall that required a possibility to be removed from the equasion.

This reminds me of another riddle. If you toss a coin, there's a 50/50 chance it could be heads or tails. If you toss it 9 times and it comes out tails all 9 times, what is the chance that the 10'th toss will come out heads. It's still 50/50 of course.

I'll bet the answer is probably related to something like the conundrum below.

3 tired sailors enter a hotel. The clerk says, "One room costs $30." The sailors agree and each gives the clerk $10.
Minutes later, the clerk remembers the room was on sale for $25. He calls the bellboy and hands him $5 saying, "the sailors overpaid. Return this $5 to them."
The sneaky bellboy, figuring the sailors weren't away of the real price, gives them $1 each and keeps $2 for himself.
So, if each sailor only paid $9 (to make $27), and the bellboy kept $2 (to make $29) what happened to the other dollar?

The ruse introduced here is the math. It is completely irrelevant to the amount of money paid out. I wouldn't be surprised of this quiz is along those same lines.

And we're back! Carry on!

Anyone that was interested in the Monty Hall problem discussion (that was here), I've split that off into it's own thread here:

http://www.wrongplanet.net/postt216080.html

Heh. Now that I've read up more on the Monty Hall thing though, I'm not so sure about my answers here now. I think they're right, but man I thought the other was too, and I wasn't.

Well it is like the Monty Hall problem. When you know the top of the coin is heads, you eliminate the coin with two tails and are left with two choices, 50/50.

I got 50-50.

If it's heads showing, you can automatically exclude the double tails coin. It can only be the double heads or the heads and tails coin.

OK so now to the real question. When I read the question I immediately knew the answer. I didn't think of it in terms of the answer in the book, I just examined the coins in my head for a second and knew. Like many of you the answer was so ridiculously obvious that I thought this must be a trick question. What really struck me was the bolded part of the quote about being counter intuitive to many people. I've been thinking about this and I still can't understand how it's counter-intuitive. In fact I am completely unable to 'see' it any other way. I wanted to wait to explain til page 2 to see if you guys would also be uncertain of your answer because it just seemed too obvious to be correct. It seems to me that most of you saw it the way that I did so my questions are as follows. Post your wisdom and thanks for being my guinea pigs

1. Is this an example of difference in perception in ASD?

2. Are there any studies you guys know of that have examined ASD vs. NT using something very similar to this type of problem(s)? I don't even know what category to search for.

3. Can anyone explain why 2/3 would be counter-intuitive and if it is counter-intuitive what other way is there to perceive the problem?

Wait... wth...
3x10=30-25=5-3=2
If I do it in reverse...
2+3+25=30
but if I multiply...
3x9=27
but 25+3=28
what just happened? this is going to bother me now...

Concerning the OP, I immediately thought 2/3 but didn't give it much more thought because I suck at probability.

Last edited by nonames on 21 Nov 2012, 11:40 pm, edited 1 time in total.

Alright, I did some testing to figure it out. I super glued pennies together to make one two headed coin, one two tailed coin, and two coins together to make a head/tails coin. Last one was so that I couldn't feel the difference as far as one coin versus two glued together.
Then, I proceeded to place the coins into a sock; then I made an excel spreadsheet to record my results in. I grabbed a coin without looking 21 times because it's divisible by three. I shook up the sock in between "grabs". It copied and pasted here fairly well.

EDIT: The formatting got messed up when I posted it. I'll be trying to fix it
EDIT: Does anyone know how to enable html tags in posts? The <pre/> tag would work if I can get it to read html.
EDIT: Alright, used the code tag for bbcode. Formatting now is correct.

Last edited by r84shi37 on 21 Nov 2012, 11:43 pm, edited 6 times in total.

Lol, for some reason I overlooked the part that says "without looking at it..."

Now that I've read it correctly, 2/3 makes perfect sense.