17 Apr 08
1 edit
Originally posted by eldragonflyWe keep going back to the same old arguments because they are the arguments which demonstrate the information you fail to take into account.
Wrong. Then you couldn't perform the card trick as stated in the original, i.e. not the wikipedia page, word problem. You start of with three cards, then the gold/gold card is *magically* eliminated, leaving only two cards to choose from, the SS and the SG, by definition. Indeed this is the foundation of this paradox.
You're diverging early on in the process, by not accounting for sides in addition to cards, and by your failure to consider all possible cases thoroughly.
Intuitive "common sense" answers can be wrong in some instances, Mathematical rigor is needed to be sure your answer holds water.
The safest approach is a systematic approach, and I have yet to see any specific point where any of the explanation either others or myself has posited derive a conclusion without a firm basis and foundation for it.
Going by baby steps again. What the heck, this time we'll not even bring the gold/gold card into it whatsoever, so that it doesn't act as a red herring to divert your attention.
Here are the questions and answers which lead to my answer and the answer of the majority of the posters in the this thread (as well as those who have added to the wikipedia article as well)
Question 1) What are the chances of each card being picked?
Answer 1) There are two cards we are looking at, the SS and the SG. Each has 50% random chance of occurring according to givens in the setup.
Question 2) What are the chances for each color facing up for each card?
Answer 2) For the SS card, it will always come up silver. For the SG, it will come up silver 50% of the time, and gold the other 50% of the time. Chances are given as fair and random according to the problem.
Question 3) What are all the possible combinations of front side & back side and how likely is each?
Answer 3) The odds for a silver front hiding a silver back is the 50% of the time the SS card is selected. The odds for a silver front hiding a gold back is 50% * 50% or 25%. The odds for a gold front hiding a silver back is 50% * 50%or 25%.
Question 4) I've discovered the side facing up is silver. How do I handle the gold-side-up cases?
Answer 4) Since gold didn't show on the front, you eliminate and scratch those cases off your list, rather than simply ignoring that it could happen, or trying to roll it into the silver front/gold back possibility. The two cases AREN'T the same.
Question 5) So how does this affect my odds?
Answer 5) You've limited the possibilities to 75% of what they could have been before the silver side was shown. If the process was repeated from scratch, the next time could be part of the remaining 25% of the time, or also be part of the 75%.
At any rate, 50% of the 75% of the time the event meets the conditions in the problem, it is the silver/silver. Another way to put it, 2 out of the 3 qualifying cases for the condition involve the SS card rather than the SG card.
There is no *magic* in eliminating the gold/gold card. It is eliminated because new information precluded the possibility that it had occurred. There is similarly no magic in reducing the chances the SG card was the one picked. New information (beyond the setup and factors leading up to the event) reduce the odds that it was the one selected, because it eliminated half the times that card would have been selected.
The SS card was not similarly reduced, and therefore when you calculate the equivalency based on the new information, the odds shift in favor of the SS card, and hence in favor of a silver backed card.
Once again I reiterate the importance of separating out the information about how the event is run, and which is learned beforehand, from the information which pertains to what actually happened and cannot be known until after the event actually occurs.
The silver face showing is a part of the RESULTS, and not a part of how the event is setup.
17 Apr 08
Originally posted by broblutoNo, Bifrost gave a sufficient proof. This is the pigeonhole principle. Here's a clearer proof:
no. it proves that it's probable. It doesn't prove that it exists. to prove that it exists, you need to get 2 people from NYC and 2 from Vatican all with 200k hairs.
Since there are only 200,001 possibilities for number of hairs on the human head (because the maximum number of hairs is 200,000 but someone can be bald), it is possible that we could select 200,001 people from NY City at random, and each one could have a different number of hairs on their head. However, if we choose 1 more person, there is no possible number of hairs that person could have that wouldn't be duplicated by one of the original 200,001. Therefore, since there are more than 200,001 people in NY City, at least 2 of them must have the same number of hairs on their heads.
This is not the case in the Vatican, which only has a population of about 800. There may in fact be 2 people with the same number of hairs on their heads in the Vatican, but you can't prove it without additional information.
17 Apr 08
Originally posted by PBE6My bad. Forgot the assumption that everyone has no more than 200k hairs.
No, Bifrost gave a sufficient proof. This is the pigeonhole principle. Here's a clearer proof:
Since there are only 200,001 possibilities for number of hairs on the human head (because the maximum number of hairs is 200,000 but someone can be bald), it is possible that we could select 200,001 people from NY City at random, and each one could have a differe ...[text shortened]... f hairs on their heads in the Vatican, but you can't prove it without additional information.
17 Apr 08
1 edit
Originally posted by geepamoogleCongratulations. You have faithfully regurgitated yet another linear, bogus and unnecessary solution/interpretation. And "mathematical rigor" would include knowing the difference between what constitutes a single event and what constitutes multiple events. 😞
We keep going back to the same old arguments because they are the arguments which demonstrate the information you fail to take into account.
You're diverging early on in the process, by not accounting for sides in addition to cards, and by your failure to consider all possible cases thoroughly.
Intuitive "common sense" answers can be wrong in som silver face showing is a part of the RESULTS, and not a part of how the event is setup.
17 Apr 08
1 edit
Originally posted by eldragonflyTell me where in my solution I have taken any steps for which I do not have a logical foundation, then. If the answer is in error, then at some point I must have made some leap and landed firmly in a chasm of illogic.
Congratulations. You have faithfully regurgitated yet another linear, bogus and unnecessary solution/interpretation. And "mathematical rigor" would include knowing the difference between what constitutes a single event and what constitutes multiple events. 😞
Show me where this supposed chasm is.
I also reject your notion that you handle single events any differently than you would handle a number of identically run events.
If you want to test odds for a single event where some information is given for the results, you should find that it will follow odds for any number of runs of the same kind of event where you ignore any result which does not match the information you have.
So even though information on results can shift the odds based on what you know, it is still the same for one-time or many-times.
Now let me define what I consider to be 'setup' and what I consider 'result', because I do draw a distinction between the two.
The information regarding how an event occurs, and the procedures governing the event are a part of setup. Setup all occurs before the event happens in time. I believe it to be fundament to examine the setup by itself, and understand what is happening before you bring into the picture information on what occurs.
It is useful to know what can happen and how likely it is BEFORE the event happens.
After the event occurs, the results for the event are set in stone. It happened in one certain way, and that cannot change. However, we often do not fully know what actually happened and where we lack information on results, we fall upon probability and the information we have on set-up.
With partial information on the results, we can eliminate some of the things which might have happened, and very well could happen if the event were repeated independent of this event.
Do note that until the event actually occurs, we have no information on the actual result. The actual result and the chances of the result are two different things.
So the procedure is to get a grasp on the odds in the most thorough manner possible, then thoroughly examine all possibilities (along with their odds) through any information we know on what actually happened, then examine what you have left to figure the equivalent odds you're looking for.
And now some practice...
I'm tired of gold and silver, so I'll go red/green this time.
I have three cards of identical feel, weight, texture, size, shape, etc. and a bag capable of holding all three. One of these cards is red on both sides, one green on both sides, and one red on one side and green on the other.
I place all three cards in the bag and mix them thoroughly for a good 5 minutes before randomly drawing one of the three cards and placing it firmly on the table without looking at which card I drew, or which side I placed downwards.
1) How likely is it I have red facing up with a red back?
2) How likely is it I have red facing up with a green back?
3) How likely is it I have green facing up with a red back?
4) How likely is it I have green facing up with a green back?
17 Apr 08
2 edits
Originally posted by geepamoogleConsider this a new problem then..
1) How likely is it I have red facing up with a red back?
2) How likely is it I have red facing up with a green back?
3) How likely is it I have green facing up with a red back?
4) How likely is it I have green facing up with a green back?
I would like to see your answers.
17 Apr 08
You refuse to answer a legitimate question? What do you fear from answering?
If the answer is related and you are correct, you ought to come out of it vindicated.
If the answer is unrelated, then the answer has no bearing.
You are probably right I should give up here. After all, what I am trying to seem would seem to be impossible.
After all, one cannot lead a blind man who refuses to accept that he cannot see.
18 Apr 08
Originally posted by eldragonflyWhy are you in this thread? Are you here to learn or here to berate?
Congratulations. You have faithfully regurgitated yet another linear, bogus and unnecessary solution/interpretation. And "mathematical rigor" would include knowing the difference between what constitutes a single event and what constitutes multiple events. 😞
18 Apr 08
1 edit
Originally posted by geepamoogleIronic, isn't it? i mean... yours can hardly be a legitimate question, you have plundered your credibility long ago.
You refuse to answer a legitimate question? What do you fear from answering?
If the answer is related and you are correct, you ought to come out of it vindicated.
If the answer is unrelated, then the answer has no bearing.
You are probably right I should give up here. After all, what I am trying to seem would seem to be impossible.
After all, one cannot lead a blind man who refuses to accept that he cannot see.