There are like six different resolutions to the Liar's Paradox. The simplest one is that all statements have an implicit "It is true that..." attached to them so the statement is really just 'A and not A' so is inherently false and not self contradictory or paradoxical.
Another resolution is to adopt a logic in which contradictions are allowed, but don't lead to absurdity.
I know there's a resolution, it's still a genuine paradox in a way that the Envelope Paradox isn't. Generally, a genuine paradox arises from a seemingly correct logical formulation actually being nonsensical within the logical framework in which it is stated. A pseudo paradox is one that appears to be paradoxical, but which is actually just an incorrect argument that can be shown to be invalid within the logic in which it is stated.
Russell's "set of sets which are not members of themselves" was a genuine paradox within naive set theory and resulted in them redrafting the axioms of sets. The various proofs that 1=2 are only pseudo paradoxes. They don't require any redrafting of the laws of arithmetic, merely finding the error.
The envelope paradox is of the same type as the proof that 1=2.
"This sentence is false" is similar to "the set of sets that are not members of themselves".
Your final sentence is correct (if X is defined as the difference between the envelopes). But so is the sentence "Switch has a 50 % chance loss of Y/2 and a 50% chance to gain Y." (Where Y is the amount in the envelope you've chosen).
Y doesn't have a constant value, so you can't use it as if it does. Its a mathematical version of equivocation. The Y you have a 50% chance to gain and the Y you have a 50% chance to lose aren't the same
Y doesn't have a constant value, so you can't use it as if it does. Its a mathematical version of equivocation. The Y you have a 50% chance to gain and the Y you have a 50% chance to lose aren't the sameSwitch has a 50 % chance loss of X, and a 50% chance to gain X.Your final sentence is correct (if X is defined as the difference between the envelopes). But so is the sentence "Switch has a 50 % chance loss of Y/2 and a 50% chance to gain Y." (Where Y is the amount in the envelope you've chosen).
You have a fifty percent chance of gaining Y and a fifty percent chance of losing Y/2.
The Y isn't changing although it is unknown. It's the amount of money in the envelope you've chosen.
Check out my coin: thttthhthtththhtthhtthtt ( <-- real, randomly generated flips )
Who wants to bet the next flip is a Head??
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Not me.
The question is, how is the reveal handled? If the reveal of who got the short straw is not until the end, then the math works out accurately. If the reveal of each straw is given as soon as it is drawn, then you're better off going last.
The question is, how is the reveal handled? If the reveal of who got the short straw is not until the end, then the math works out accurately. If the reveal of each straw is given as soon as it is drawn, then you're better off going last.The reveal doesn't change a thing.
It changes your likelihood of drawing a straw at all, which in turn alters your tactical viewpoint of the game. While the percentage chance of you drawing the short straw is the same, the ultimate outcome is that you will draw straws at all the fewest amount of times.
Played right, it can give a psychological edge that can change the choices of others and potentially skew the results more in your favor.
Also, if you're drawing last, you're usually the one holding the straws, so you typically already know which one is the shortest.
Played right, it can give a psychological edge that can change the choices of others and potentially skew the results more in your favor.But not the odds.
The choices of others are your odds.
The choices of others are your odds.No, the choices don't actually affect your odds. They just determine the results.
Not mathematically true. Especially since odds exist to predict the chance of results coming about.
Let's take the math Steve Geddes posted on the first page and alter it to reflect the the first player drawing the draw one more game on average due to the manipulations.
2/4 x 2/3 x 1/2 = 4/24 = 1/6
That is the approximate chance, at the end, of the final player drawing a straw, just due to the first player drawing the short straw more frequently. It is not altering how many straws are actually short straws, but is altering how frequently the first player draws the straw.
Yes. If you magically change the odds so the first person is more likely to draw the short straw, it's better not to be the first person. Still doesn't matter whether you're second or last, unless you also decide to change those odds.The choices of others are your odds.No, the choices don't actually affect your odds. They just determine the results.Not mathematically true. Especially since odds exist to predict the chance of results coming about.
Let's take the math Steve Geddes posted on the first page and alter it to reflect the the first player drawing the draw one more game on average due to the manipulations.
2/4 x 2/3 x 1/2 = 4/24 = 1/6
That is the approximate chance, at the end, of the final player drawing a straw, just due to the first player drawing the short straw more frequently. It is not altering how many straws are actually short straws, but is altering how frequently the first player draws the straw.
The math posted was used by Geddes to show that the odds of the first, second, and third person on their individual picks do not change the overall odds of the last person picking the short straw.
I'm demonstrating that changing the rate of short straw picks of the first person through manipulation does change the odds of the last person picking. Which, as you noted, can be cheating in some of the ways it's done.
Yes. If you magically change the odds so the first person is more likely to draw the short straw, it's better not to be the first person. Still doesn't matter whether you're second or last, unless you also decide to change those odds.The choices of others are your odds.No, the choices don't actually affect your odds. They just determine the results.Not mathematically true. Especially since odds exist to predict the chance of results coming about.
Let's take the math Steve Geddes posted on the first page and alter it to reflect the the first player drawing the draw one more game on average due to the manipulations.
2/4 x 2/3 x 1/2 = 4/24 = 1/6
That is the approximate chance, at the end, of the final player drawing a straw, just due to the first player drawing the short straw more frequently. It is not altering how many straws are actually short straws, but is altering how frequently the first player draws the straw.
The math posted was used by Geddes to show that the odds of the first, second, and third person on their individual picks do not change the overall odds of the last person picking the short straw.
I'm demonstrating that changing the rate of short straw picks of the first person through manipulation does change the odds of the last person picking. Which, as you noted, can be cheating in some of the ways it's done.
This is only true if you are allowed to change the draw order on the fly AND you are the ONLY one allowed to change the order.
Basically, you are describing a method of leveraging the unfair advantage you already had.
Nah. It just requires some proper social engineering. Volunteer to hold the straws and be the last pick, keep the position, and then just use psychological tricks to get whoever is first to draw the short straw half the time.
You don't change the order of the straws, don't change the draw order at all, and you just hope someone else doesn't decide to take over.
*facepalm*
Yes. It's in the post you first replied to.
Tone of post is not sarcastic; it's amused. I'm sitting here laughing at myself.
Subtle facial clues that hint one straw is the short one is my preferred go-to. Switching up whether or not it's the short straw I'm hinting about after awhile of hinting at the same type is also good; causes them to misread and pull the wrong straw. Especially if it only happens every so often...
I have also used this to end up with the short straw a few times. There are times when the group insists upon drawing straws for an event, but you know you have to be the one to do it.
Subtle facial clues that hint one straw is the short one is my preferred go-to. Switching up whether or not it's the short straw I'm hinting about after awhile of hinting at the same type is also good; causes them to misread and pull the wrong straw. Especially if it only happens every so often...
I have also used this to end up with the short straw a few times. There are times when the group insists upon drawing straws for an event, but you know you have to be the one to do it.
Again, you are describing ways to leverage your already unfair advantage. That's not the topic at hand.
Again, you are describing ways to leverage your already unfair advantage. That's not the topic at hand.Subtle facial clues that hint one straw is the short one is my preferred go-to. Switching up whether or not it's the short straw I'm hinting about after awhile of hinting at the same type is also good; causes them to misread and pull the wrong straw. Especially if it only happens every so often...
I have also used this to end up with the short straw a few times. There are times when the group insists upon drawing straws for an event, but you know you have to be the one to do it.
The topic at hand is if it is better to go first or last when drawing straws (which is what I addressed in my first post).
The mathematical side of a purely-random drawing was covered. I'm covering the human element that exists outside of the mathematical element and which may involve items that influence probability or outright exist outside of it, specifically the possibility that one of the people involved may have an unfair advantage. Which is something necessary to take into consideration, but not taken into consideration all of the time.
While mathematics is nice for figuring out purely random outcomes, there is also the human element to consider, and it is ultimately the human element that has the most effect. Knowing, for example, that a friend always draws the third straw from the left means that friend can be safely eliminated by making certain that straw is the outcome you wish that friend to deal with, despite the fact the outcome does not match a purely random mathematical output in results. And even if there is no cheating going on, knowing that fact about that friend and being the one holding the straws means you already know the outcome of their choice.
If you know absolutely no one will cheat, you can take a look at the math presented on this thread and see your chance of ending up with the short straw at any opportunity to choose clear back on page 1. If you do not know that or you know one person will cheat (or several people will cheat), you have to start considering nonrandom outcomes and probability equations that do not produce purely random chance.
In short, I was presenting the nonmathematical aspects that must be considered as part of the decision making on placement in choosing straws.
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The (unstated) assumption is that the game is fair and nobody is cheating. However, cheats do not "exist outside" of either probability or "the mathematical element". You can mathematically account for cheats too.
For example: if you (somehow) know another player is going to cheat, the best you can do to maximise your chances is to do whatever you can to go before that person makes their choice (although, in my view, this in itself constitutes cheating on your part). Knowing another player's proclivities (like third from the left or something) doesn't actually help you at all unless you plan on cheating yourself.
Obviously one answer to "how do I maximise my chances of winning this game of chance?" is always going to be "Cheat". It's not generally a very useful answer though, in terms of advancing understanding, since everyone already knows that answer.
I've always liked this one:
Coupons in cereal boxes are numbered 1 to 5 and a complete set, one of each, is required to get a prize.
With one coupon per box, how many boxes on the average are required to make a complete set?
I also like this question, because it reminds me of "Harmonic Numbers" and "Recursion". Good stuff.
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