Probabilty Riddles

[uphigh_downlow]

I feel really stupid having even attempted the math. This is such an elegant and simple reasoning.

Sometimes the easiest things are right infront of your eyes, and you cant see them.

Thanks for openingg my eyes [img]/images/graemlins/smile.gif[/img]

[Thythe]

Yeah I didn't see that either. Very clever.

[Siegmund]

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It is perhaps simpler and key just to acknowledge that both jars finished with the same volume of liquid they started with.Since the liquids are conserved between the jars, whatever one jar lost the other jar gained. Whatever they lost must have been replaced by an equal volume from the other jar - the jars simply exchange a volume of eachothers liquid which muist be the same since, they both end up with the the same volume they started with.


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Yes.

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EG If yhe vinegar jar has 75% vinegar left then it must have 25% oil. So the oil jar must have 75oil, 25% vinegar.

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And no.

The result applies to absolute quantity in the wrong jar, NOT to percentage. The percentages will only be equal if both jars have the same volume (which was not specified in the original post.)

[gaming_mouse]

nice.

[chaosuk]

'And no.'

A fair cop, but it was 4am. Though I thought the problem was a mixture problem , which, as you said, required equal sizes and which is, of course, a special case of the problem amount.

I only just caught end of the thread and hence the answer. No doubt had this been an exam question, I'd have gone down the same maths route, but since I had the answer (& the maths had been done) I could look for a more simpler explantion. It is particularly disturbing, though, how we've trained our minds to approach these problems - a solution that involves simple addition and subtraction and one that a small child could understand is not immediately apparent.

[stankphish]

"There is a jar of oil and a jar of vinegar. One spoonful of vinegar is poured into the oil. A spoonful is then taken back out of the oil-vinegar mixture, and poured back into the vinegar?

Is there now more vinegar in the oil, more oil in the vinegar, an equal amount of each in the other, or is it impossible to tell?"

I think im missing something but. When you put a teaspoon of vinegar into the oil and then you put a teaspoon of the oil and vinegar back into the oil.

Dosnt that mean that you are putting more vinegar into the oil b/c you are putting oil and vinegar back into the vinegar meaning that there is more vinegar in the oil?

I understand the logic that they both have the same amount of liquid that they started with but thats not what the problem asks. It asks which jar has a larger amount of the other substance in it. Hypothetically if you spooned out of jar with oil and vinegar in it and the spoon contained 3/4 oil and 1/4 vinegar and you put that back in the vinegar jar there would be 3/4 spoonfull of vinegar in the oil and 3/4 oil in the vinegar... ok i just figured it out

[OrangeKing]

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For question 1.
You either picked the right door off the bat or you didn't
Probablity that you picked the right door is 1/3 so the chances that door B is correct is 1/3. Conversly the probability that you picked the wrong door is 2/3 so the chances that the other door is correct is 2/3.

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Huh? The new information tells you that your choice has 50% chance of being correct so switching does you no good. The answer quoted (switch doors) must be a silly play on words.

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The problem is that the stated problem in this thread isn't EXACTLY like the Monty Hall problem. The missing bit of information is:

You know the host will always open a wrong door every time you play this game. So it's not a surprise when he shows you one of the losing doors.

If he just decides to show you a door for no reason, you've gained no information. But when you know ahead of time that he's going to show you a losing door no matter what you pick...then you've gained a lot of information.

Same goes for the 100 doors example. If you know he'll show you 98 of the losing doors after you pick your door, do you really think it makes no difference whether you stick with your original door, or switch to the one other door he didn't open? Of course not; there's a 99% chance that other door is the winner. On the other hand, if you and 99 other people each have a door, and you see 98 people open their doors and lose, you and the other remaining contestant each have a 50% chance of winning.

[PairTheBoard]

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equal amounts in each, if you start with equal volumes in the jars

Xo and Xv to start X is the # spoonfullss in each

Xo + 1v and (X-1)v after the first exchange

(Xo + 1v){1- (1/X+1)} and (X-1)v + (Xo +1v)/X+1
after second exchange

Turns out that there is X/(X+1) of one in the other, both ways

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Your answer is correct, but I believe this calculation is assuming that they are getting evenly mixed, which you cannot assume. That's why the question uses oil and vinegar. In fact, you cannot know how much of each will be in the other -- there might be 0, and there might be a full spoonful. But even so, you CAN always know that the amount of contamination will be equal. There's a simple logical argument, which requires no algebra, and is the reason I like the puzzle.

gm

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Right. All you have to notice is that after exchanging teaspoons the volumes are as before so whatever oil is in the vineger is missing from the oil and must therefore have been replaced by an equal amount of vineger.

PairTheBoard

[theTourne]

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I always found it easier to think of that question using 100 doors-

You pick one-
The host opens 98 empty ones-
Should you keep the one you originaly picked or the remaining unknown door?

Much easier to grasp the concept that way.

regards,
Tim

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A guy and I were trying to explain this concept to a friend of mine. We used your approach, and said "pretend there are 100 doors, he's going to show you 98 empty doors," etc. Well my friend was still confused, and after a few more agonizing explanation attempts the other guys said, "OK, well then imagine that you have a million doors..."

It was hilarious at the time.

[DonkeyChip]

Hi all, first post here.

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You know the host will always open a wrong door every time you play this game. So it's not a surprise when he shows you one of the losing doors.

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OK, NOW I get it. I thought you guys were crazy. That's VERY important!

Think of it another way (under the original premise that the second door opened is RANDOMLY chosen): Suppose you originally picked the correct door; in that case then EVERY time Monty opens another door (randomly), it will be empty. But if you originally chose incorrectly, then only 50% of the time Monty RANDOMLY opens another door it will be empty. So once Monty opens that second door and it's empty, then it's TWICE as likely that your original choice was correct. So using that logic, you should STICK with your original choice. But this logic is just as flawed as the original logic (provided we think that the second door is RANDOMLY chosen).

If that's a bunch of horse-hockey, lemme know. [img]/images/graemlins/smirk.gif[/img]

Regarding the vinegar and oil: Vinegar and oil for the most part don't mix and vinegar is more dense than oil, so the teaspoon of vinegar just sinks to the bottom of the oil. So you've put 1 teaspoon of vinegar in the oil and then put 1 teaspoon of oil into the vinegar. There are equal amounts of each in the other. They could've chosen a lot of different liquids/substances but I'm assuming they chose oil and vinegar for a reason.

I like these kinds of riddles, gives me something to think about while driving to/from work.