This question first appeared on this blog a couple of years back.
Check out my 10th July 2005 post:
"Five Clues - One for each 10 of 50"
The question was this...
You have three cards, (two aces and a queen), all face down. You are asked to find the queen, and you are asked to pick one card only. Then, the magician looks at them all, and promptly turns face up an ace. What is the probability that your card is the queen?
The answer is a third, since you made the choice when all three were face down.
NOW consider the same scenario...
You have three cards, (two aces and a queen), all face down. You are asked to find the queen, and you are asked to pick one card only. Then, the magician looks at them all, and promptly turns face up, an ace.
Your delicate finger is still on the card you chose before any were revealed, and YOU KNOW the probability is 1/3 that yours is the queen.
So, you have 3 cards, two face down, and an ace face up. You have tiny finger on your face down card, knowing that it is 1/3 that it is the one queen.
Now, lift you finger up, really carefully. You have 3 cards, one face up, an ace, and two face down, the other ace and the queen. You are then asked to make your choice again, so you put your same finger down, ever more carefully, ON THE SAME CARD YOU CHOSE BEFORE.
NOW, what is the probability that your card is the queen?
The astonishing answer is that it is now 1/2.
So....what changed between your lifting your finger off the card, to replacing it back, to cause this change in probability?
Moreover, is the answer different if you had chosen the other card instead?
Comments welcome.
Friday, May 02, 2008
Find the Queen..revisited.
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6 comments:
For the first question, well, is changed the number of possible events.
For the second one, the answer would have been the same (1/2) and this because the two events are indipendent (equal probability) and there is a queen so,
eventA(I chose the queen) + eventB(I don't chose the queen)=1(there is a queen)
but even
eventA=eventB (equal probability)
ciao
Yes, the probabilities have already been given.
However, I asked what changed between when your finger was raised, and when you put it down again.
It is a very easy question to answer and does not involve calculation.
You are nearly there, you only need to specify further.
Okay, without calculation :) ,
that is changed is the magician give me another chance to choice my card and the second time the possible events are only two (instead of three of the first time).
It seems nearly to be at school, good sensation :D
ciao
Si Simone,hai ragione. When there are many distinct possibilities, the 'possible outcome' is in a 'mixed state' before it's realized. The more the number of distinct possibilities, the less the probability for a particular outcome to occur. So, the probability of a particular outcome is dependent on our 'knowledge' of the system. To demonstrate this, let me now give a simple example:
There are four boxes A,B,C, and D , one marble and three persons 1,2,3. The marble is in one of the four boxes. 1 knows that the marble is NOT in box A , 2 knows that it's NOT in A and B. 3 doesn't know anything about what box the marble is in. Therefore, for 1 , the probability of finding the marble in the first draw is 1/3, for 2, it's 1/2 and for 3 it's 1/4. As we can see from this simple example, the knowledge of the system changes the probability of the outcome.
Without any knowledge about the system described above, Quantum Mechanics says that the existence of the marble is 'fuzzy' i.e, it exists in ALL the four boxes simultaneously! It's only as we start looking into the boxes one by one that this 'fuzziness' gradually reduces until it totally disappears when the marble is found! Technically, this disapperance of the quantum 'fuzziness'
is known as Wave-function collapse, and the 'fuzziness' is called the mixed-state.
I would love to see a post here about entropy and wave-function collapse so we could discuss more and more exciting stuff. Thank you Aki and Simone for giving me an opportunity to discuss here.
The marble's existence is NOT fuzzy, potato, because it is a MARBLE, being of macroscopic size (macroscopic meaning the size is much greater than Planck's constant). Your argument is irrelevant at best, and misleading at worst. Quantum mechanics is only relevant to stuff available that we cannot "see" with the naked eye, and definitely does NOT apply to marbles, cars, tennis balls, cats, etc.
To claim that "everything" is governed by Quantum Mechanics is like saying "Our basketball team should do well on the planet "oohhuhuhooompha" despite the fact that we have never played them, played there, and have never played basketball at all in seven dimensions".
The point being?
Quantum Merchanics is only relevant at very small scales.
Might Bayesian Statisitics be applied to either the card trick or the marble idea? I think it was Polya or someone like that who said one of the first questions one must ask when attempting to solve a problem is: "Is there sufficient information available to solve the problem?"
Your answer is different based on your different choice, to answer the latter question, in my humble opinion.
What changed with the lifting of the finger is you ended the first problem and took up a second problem, so to say what is different I am tempted to say nothing, but in fact you have chosen to play a different game that was somewhat dependent on the first game. To me the first game ended without resolution, but the resolution to the second game would have also been the resolution to the first game. In theory, I believe they have no connection, though.
In other words, I find this scenario to be misleading from the start with the intention of concluding with the opposite answer of whatever is suggested by the person answering the question. I believe Magician's use such tricks for entertainment purposes and to hold an audience.
So, what is your answer to this whole post? Okay, maybe I am bored and not paying enough attention. ;-) I will try harder the next time I read this. Yawn!
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