For İntermediate:Edgy Grammar and Reason - Dini ilimler, Eğitim ve Kişisel gelişim siteniz

**Tunaltay** · 01-15-2007, 12:44

Introduction:

"EST. TIME: 30 MINS PLUS. Human reasoning and language comprehension are closely related capacities. This quiz plays in the area where they most clearly intertwine. No factual knowledge is tested here, 'just' analytic reasoning and language processing."

Question 1:

If no head injury is too trivial to be neglected, then:

- if you injure your head, consult a physician.
- if you don't consult a physician, your head injury must be trivial.
- don't injure your head unless you consult a physician.
- don't consult a physician no matter how trivial your head injury may be.

Question 2:

Only one of the following sentences cannot be disambiguated to parse as a grammatical, meaningful sentence in English, using standard grammatical rules. Which one is the agrammatical sentence?

- Buffalo buffalo buffalo buffalo.
- The horse raced past the barn fell jumped.
- The rat the cat the dog chases hunts hides.
- The claim the horse he entered was a ringer was false.

Question 3:

Consider this simple syllogism. '(1) All artists are beekeepers. (2) Some beekeepers are chemists.' If (1) and (2) are both true, then which of the following must be true?

- Some artists are chemists.
- None of the other options is correct.
- Some chemists are artists.
- Some chemists are not artists.

Question 4:

Three cards are in a hat. One is red on both sides, one is white on both sides, and one is red on one side and white on the other. I draw a card from the hat, and drop it on the table. The upward-facing side is red. What are the odds that the downward-facing side is also red?

- Fifty percent.
- Two out of three.
- One out of three.
- None of the other options is correct.

Question 5:

Tom, Dick and Harry are in prison. One of them has been randomly selected to die in the morning, and the other two will be set free. Their guard knows which one will die, but none of the prisoners does. The guard is under strict instructions not to divulge the identity of the doomed man. Tom is desperate for any information beyond the fact that his odds of death are one in three. He begs the guard to throw him an informational bone. Finally, to shut him up, the guard agrees to reveal only the following: the name of one of Tom's fellow prisoners who will be set free rather than killed. The guard then says that Dick will be set free. After receiving this information from the guard, what is the most accurate calculation Tom can make of the probability that he is the doomed man?

None of the other options is correct.

One out of two.

One out of three.

Two out of three.

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Results for Edgy Grammar and Reason

Question 1: If no head injury is too trivial to be neglected, then:
The correct answer is don't consult a physician no matter how trivial your head injury may be..

Pragmatic cues are powerful guides to how language should be interpreted, often more powerful than mere niceties of grammar. Actually parsing the sentence yields a meaning contrary to that implied by pragmatic cues or, specifically, here, what I refer to as 'keyword heuristics.'
11% of players have answered correctly.

Question 2: Only one of the following sentences cannot be disambiguated to parse as a grammatical, meaningful sentence in English, using standard grammatical rules. Which one is the agrammatical sentence?

The correct answer is The horse raced past the barn fell jumped..

'The horse raced past the barn fell' is a famous garden-path sentence, where 'raced past the barn' is used adjectivally (with respect to 'horse') rather than as a verb phrase. But with 'jumped' at the end it stops making sense.

'Buffalo buffalo buffalo buffalo' implies that bison from a particular city in New York tend to bewilder, deceive or confuse (other) bison.

The rat-cat-dog sentence is hard to explain without a whiteboard or a tree-diagram, but try to follow this: (i) 'the cat' is a noun phrase modified by the adjectival phrase 'the dog chases', (ii) 'the cat the dog chases' is the noun phrase which is the subject of the verb 'hunts', whose object is 'the rat', (iii) 'the rat the cat the dog chases hunts' is the noun phrase which is the subject of the verb 'hides', which has no object.

The horse-race-ringer-false sentence can be clarified with the following technically superfluous but practically much-needed punctuation, as follows: 'The claim ([that] the horse--[the one] he entered [in the race]--was a ringer) was false.'

[Apologies to non-native English speakers, who may have had unwarranted trouble with the buffalo sentence, which is really a vocabulary issue. Nevertheless, the correct answer (the actually agrammatical sentence) cannot be parsed, so you "should" have been able to identify it as the correct answer on pure grounds of grammar, even if you could not eliminate the buffalo sentence as demostrably wrong.]
22% of players have answered correctly.

Question 3: Consider this simple syllogism. '(1) All artists are beekeepers. (2) Some beekeepers are chemists.' If (1) and (2) are both true, then which of the following must be true?

The correct answer is None of the other options is correct..

[Adding to my explanation for this problem about five years after I first posted it to this site, I note that a (to me) very surprising number of people seem to get this wrong. At time of writing, quiztakers have performed at chance levels on this problem. Although only a handful of people have written to me angrily about this one, unlike problems 4 and 5, which still get me frequent hate-mail, despite all my efforts to ward it off (see below).]

It is possible that all of the beekeepers who are chemists are also artists. It is possible that none of the beekeepers who are chemists are also artists. There is no justifiable conclusion about the overlap or non-overlap of chemists and artists.

[The paragraph immediately above is my first stab at explaining the problem; let me now expand that explanation a little further.]

Let us use the following convention: an 'A' is an artist, a 'B' is a beekeeper, a 'C' is a chemist, an 'AB' is an artist who is also a beekeeper, a 'BC' is a beekeeper who is also a chemist, an 'ABC' is an artist who is also a beekeeper and a chemist, and so on.

Here is one set that satisfies both premises of the syllogism: AB, AB, ABC, ABC. In this set, every A is also a B (premise 1), and some Bs are also Cs (premise 2). In addition, some As are also Cs, and all Cs are also As.

But here is another set that also satisfies both premises of the syllogism: AB, AB, ABC, ABC, BC, BC. Here, too, every A is also a B (premise 1), and some Bs are also Cs (premise 2). But here, while some As are Cs, as above, this time some Cs are not As.

And here is yet another set that also satisfies both premises of the syllogism: AB, AB, B, B, BC, BC. Again, every A is also a B (premise 1), and some Bs are also Cs (premise 2). But in this set, no A is also a C (and no C is also an A).

Because each of these possible sets (among many others) satisfies both premises of the syllogism, it is not justified to reach any conclusion about the relationship between Cs and As: some, all or none of the Cs could be As, and some, all, or none of the As could be Cs.

I hope this helps!

[Writing now more like 6 years on from originally posting this problem. People have started writing to me saying things like, "I JUST KNOW that some of the artists are chemists; I can't prove it, but I JUST KNOW your logic is wrong." PLEASE DO NOT WRITE SUCH A MESSAGE TO ME. In the absence of the additional premise "All beekeepers are also artists" the conclusion "Some artists are chemists" is unjustified. It might be true, but it might not be true. If it might not be true, you CANNOT conclude it "must" be true.]
27% of players have answered correctly.

Question 4: Three cards are in a hat. One is red on both sides, one is white on both sides, and one is red on one side and white on the other. I draw a card from the hat, and drop it on the table. The upward-facing side is red. What are the odds that the downward-facing side is also red?

The correct answer is Two out of three..

The problem uses three cards (R/R, R/W, W/W), each with two sides. Each time a card is drawn there is thus a 1-in-3 chance that any particular card will be drawn, but from a formal standpoint that is a red herring, because, relevantly, there is a 1-in-6 chance that any particular SIDE will come up. That is, there is a 1-in-3 chance that the R/R card will be drawn, and then, given that R/R has been drawn, there is a fifty-fifty chance that the "A" side will come up and a fifty-fifty chance that the "B" side will come up. (The two sides are functionally identical, but, again, that fact is a red herring from a formal standpoint.) Let us call side "A" of the R/R card R/R:A and side "B" of that card R/R:B.

It is helpful to look at this as though we were making a series of observations of the problem, as though it were an experiment. Suppose we "ran" the problem 600 times. (We would expect to see red come up 300 times and white come up 300 times, but that observation would be meaningless from the formal standpoint. Nevertheless, it is probably this fact that triggers many people's incorrect intuitions about the problem.) Relevantly, we would expect the following: R/R to be drawn 200 times (of which we would expect R/R:A 100 times and R/R:B 100 times); R/W to be drawn 200 times (of which we would expect the red side 100 times and the white side 100 times); and W/W to be drawn 200 times (of which we would expect W/W:A 100 times and W/W:B 100 times).

So the problem tells us that on this particular instance a red side came up, meaning that we can throw out all observations where a white side came up. We are left with: the 100 times we saw R/R:A, the 100 times we saw R/R:B and the 100 times we saw R/W:A. Of the approximately 300 draws culminating in a red side, approximately 200 of them would have been draws of the R/R card and 100 would have been draws of the R/W card.

Thus, if we continued to "run" the "experiment," and every time we saw a red side come up we predicted that we had drawn the R/R card, the prediction would be accurate approximately 2/3 of the time. That is to say, there is a 2/3 probability on any draw that there will be red on the other side of any red draw.

***

If you still don't understand: before you write to me about this, please, please, try this simple experiment at home. I mean really do this before you write to me. Please.

Take 3 quarters, and with a magic marker, write (i) an R on both sides of one of the quarters, (ii) a W on both sides of one, and (iii) an R on one side and a W on the other side of the third quarter. (Technically, the W/W quarter is wholly superfluous, but let's set this up just like the problem in the quiz question. However, if you really see why we don't need to use the W/W quarter, feel free to use just the R/R and R/W quarters.)

Also take a piece of paper, and write the numbers from 1 to 30 in a column, with a blank space to the right of each number.

Okay, put the quarters in a hat, and pull one out at random with your eyes closed, placing it flat on a table top before you open your eyes. The quiz problem says that red is facing up, so if you get a W, put the quarter back in the hat and repeat.

When you have an R facing up, check the other side. Write down what was on the other side (R or W) in the blank corresponding to the number 1 on your piece of paper. Repeat, writing down the result for the downward-facing side in blank number two, and continue repeating (always throwing the coin back and NOT recording the downward-facing side whenever a W is the upward-facing side) until you have all 30 blanks filled.

You will have written R approximately 20 times, and W approximately 10 times.

***

[IF YOU GOT THIS QUESTION WRONG, AND HAVE TROUBLE UNDERSTANDING HOW YOUR ANSWER COULD POSSIBLY BE WRONG, YOU ARE IN GOOD COMPANY: THIS PROBLEM IS WELL STUDIED BY PSYCHOLOGISTS, LOGICIANS, AND GAME THEORISTS, AND IT IS WELL DOCUMENTED THAT EVEN VERY WELL-EDUCATED PEOPLE GET THIS PROBLEM WRONG MORE OFTEN THAN NOT, AND HAVE TROUBLE UNDERSTANDING THE CORRECT ANSWER. IT IS HIGHLY COUNTER-INTUITIVE.

IF YOU GOT THE PROBLEM WRONG AND FIND YOURSELF TEMPTED TO CONTACT ME TO TELL ME WHY MY ANSWER IS WRONG AND YOURS IS RIGHT, YOU ARE ONE OF A BIG, BIG CROWD. BUT PLEASE READ ON BEFORE YOU DO SO.

FIRST: MY ANSWER IS SIMPLY NOT IN ERROR. THE PROBLEM HAS BEEN EXHAUSTIVELY ANALYZED BY SCHOLARS. IF YOU THINK THAT YOU HAVE UNCOVERED AN ERROR HERETOFORE UNDETECTED BY ANY OF THE MANY ACADEMICS WHO HAVE BUILT CAREERS TRYING TO UNDERSTAND HOW HUMANS REASON OVER PROBLEMS LIKE THIS ONE, THEN I BESEECH YOU, IN THE BOWELS OF CHRIST, CONSIDER THAT YOU MIGHT BE WRONG.

SECOND: THE ABOVE IS MY BEST EFFORT IN THIS CONTEXT TO EXPLAIN THE CORRECT ANSWER. I DON'T WANT TO BE CONTACTED BY ANYONE WHO CANNOT HONESTLY SAY THAT THEY HAVE SPENT AT LEAST HALF AN HOUR WORKING THROUGH MY EXPLANATION AND TESTING IT BY SIMULATING THE PROBLEM. DON'T WASTE MY TIME WITH A MESSAGE THAT, AT BEST, FAILS TO TAKE INTO ACCOUNT THE EXPLANATION I HAVE PROVIDED.

FINALLY, BECAUSE THE ABOVE IS MY BEST EFFORT TO EXPLAIN THE CORRECT ANSWER TO THE PROBLEM AS IT IS DRAFTED, I REALLY DON'T HAVE A LOT TO ADD TO IT, GENERALLY SPEAKING. MAYBE YOU HAVE A SPECIFIC QUESTION THAT, IF ASKED CIVILLY, I WILL TRY TO ANSWER. BUT IF YOU REALLY CARE ABOUT UNDERSTANDING THE PROBLEM, AND YOU CAN'T GET THERE AFTER A GOOD FAITH EFFORT TO WORK THROUGH MY EXPLANATION, I ADVISE YOU TO LOOK UP ONE OF THE NUMEROUS BOOKS ON THE SUBJECT. HOWARD MARGOLIS HAS SEVERAL BOOKS ADDRESSING THIS PROBLEM, BUT THERE ARE NUMEROUS OTHER AUTHORS AS WELL (YOU DON'T NEED TO RELY ON "MY" EXPERT!).]
18% of players have answered correctly.

Question 5: Tom, Dick and Harry are in prison. One of them has been randomly selected to die in the morning, and the other two will be set free. Their guard knows which one will die, but none of the prisoners does. The guard is under strict instructions not to divulge the identity of the doomed man. Tom is desperate for any information beyond the fact that his odds of death are one in three. He begs the guard to throw him an informational bone. Finally, to shut him up, the guard agrees to reveal only the following: the name of one of Tom's fellow prisoners who will be set free rather than killed. The guard then says that Dick will be set free. After receiving this information from the guard, what is the most accurate calculation Tom can make of the probability that he is the doomed man?

The correct answer is One out of three..

This problem is tricky for reasons not well understood by psychologists. Some characteristic of human reasoning makes it difficult to grasp the logic, even once it has been laid out openly.

I find that providing an analogous illustration helps facilitate people's understanding of this problem. Assume that there are 100 children and 100 boxes. Each box has a different one of the children's names on it. I pick one at random to put a gold star inside; inside all the rest I put a silver star. I then seal all the boxes.

Now suppose that Tom is one of the children. Because I selected the box at random, as far as he knows there is a 1% chance that he received the gold star in his box. That is, if we did this whole "experiment" 100 times, in approximately 1 run of the experiment Tom would get the gold star and the other 99 boxes would contain silver stars; and in approximately 99 runs of the experiment Tom would get a silver star, 98 other boxes would contain a silver star, and 1 other box would contain a gold star. Note that in all 100 runs, at least 98 boxes other than Tom's box contain silver stars; that will be true no matter where the gold star is.

Of course, _I_ know which box contains the gold star. I can therefore always generate a list of 98 names (always excluding Tom's name) that correspond to boxes containing silver stars. I can do this if Tom's box contains the gold star and I can do this even if Tom's box does not contain the gold star.

Suppose that I compile such a list for Tom's benefit, having assured him in advance that the list will contain precisely 98 names of persons other than Tom whose boxes contain silver stars; in other words, that no matter which box contains the gold star the list will reduce the candidate gold-star boxes to two, Tom's and one other.

HAVE TOM'S ODDS OF RECEIVING THE GOLD STAR ROCKETED FROM 1% TO 50%? CONSIDER THAT IF WE "RUN" THE "EXPERIMENT" 100 TIMES, WE ONLY EXPECT TOM TO RECEIVE THE GOLD STAR ONCE, AND YET I CAN PRODUCE A LIST OF 98 NON-TOM SILVER-STAR NAMES IN _EVERY SINGLE RUN_. WILL GENERATION OF A 98 NON-TOM SILVER-STAR-NAME LIST CAUSE TOM TO RECEIVE THE GOLD STAR FIFTY PERCENT OF THE TIME? OBVIOUSLY NOT; I CAN GENERATE SUCH A LIST WITHOUT AFFECTING THE ODDS. TOM STILL ONLY HAS A 1% CHANCE OF RECEIVING THE GOLD STAR ON ANY PARTICULAR "RUN."

I hope the correspondence to the Prisoner Problem is obvious. Change the number of boxes from 100 to 3, and the gold star to a death warrant, and the logic is identical across the two situations. The information Tom was given had no effect on his own odds.

Note that although Tom's odds remain 1 in 3, Harry's odds of being the one to be killed can now be rated at 2 in 3. This is because of the conditions the guard imposed: he said he was generating a one-person non-Tom list of people not being killed; he didn't promise to make a non-Harry list, so the new information that, of persons who are not Tom, Dick is safe doubles the probability that Harry is going down.

***

Here is ANOTHER way of explaining it. You can set this up and try it at home. In fact, please REALLY DO try this at home before you write to me.

Pick three friends to help you, and you be the jailer. Pick one of your friends to be "Tom." Put your friends' names in a hat, and pick one of them at random: that one will be "killed." Tell Tom the name of one of the other two friends who will not be killed. Ask Tom what his odds of being killed are. He will probably say 50%. Reveal to him who will actually be killed. Write down the name of the person to be killed. Do this 30 times.

It may be that Tom always thinks he has a 50-50 chance after the candidates for being killed are narrowed down to two people, but in fact, at the end of the day, you will have written Tom's name down only approximately 10 times (1/3 of 30). And of the approximately 15 times that you said "Harry will not be killed" you will have written down "Dick was killed" approximately 10 times (2/3 of 15), and vice versa. All get killed approximately the same number of times over the long haul, but only the calculation of odds for the non-Tom prisoners change as a result of the guard's information.

***

[IF YOU GOT THIS QUESTION WRONG, AND HAVE TROUBLE UNDERSTANDING HOW YOUR ANSWER COULD POSSIBLY BE WRONG, YOU ARE IN GOOD COMPANY: THIS PROBLEM IS WELL STUDIED BY PSYCHOLOGISTS, LOGICIANS, AND GAME THEORISTS, AND IT IS WELL DOCUMENTED THAT EVEN VERY WELL-EDUCATED PEOPLE GET THIS PROBLEM WRONG MORE OFTEN THAN NOT, AND HAVE TROUBLE UNDERSTANDING THE CORRECT ANSWER. IT IS HIGHLY COUNTER-INTUITIVE.

IF YOU GOT THE PROBLEM WRONG AND FIND YOURSELF TEMPTED TO CONTACT ME TO TELL ME WHY MY ANSWER IS WRONG AND YOURS IS RIGHT, YOU ARE ONE OF A BIG, BIG CROWD. BUT PLEASE READ ON BEFORE YOU DO SO.

FIRST: MY ANSWER IS SIMPLY NOT IN ERROR. THE PROBLEM HAS BEEN EXHAUSTIVELY ANALYZED BY SCHOLARS. IF YOU THINK THAT YOU HAVE UNCOVERED AN ERROR HERETOFORE UNDETECTED BY ANY OF THE MANY ACADEMICS WHO HAVE BUILT CAREERS TRYING TO UNDERSTAND HOW HUMANS REASON OVER PROBLEMS LIKE THIS ONE, THEN I BESEECH YOU, IN THE BOWELS OF CHRIST, CONSIDER THAT YOU MIGHT BE WRONG.

SECOND: THE ABOVE IS MY BEST EFFORT IN THIS CONTEXT TO EXPLAIN THE CORRECT ANSWER. I DON'T WANT TO BE CONTACTED BY ANYONE WHO CANNOT HONESTLY SAY THAT THEY HAVE SPENT AT LEAST HALF AN HOUR WORKING THROUGH MY EXPLANATION AND TESTING IT BY SIMULATING THE PROBLEM. DON'T WASTE MY TIME WITH A MESSAGE THAT, AT BEST, FAILS TO TAKE INTO ACCOUNT THE EXPLANATION I HAVE PROVIDED.

FINALLY, BECAUSE THE ABOVE IS MY BEST EFFORT TO EXPLAIN THE CORRECT ANSWER TO THE PROBLEM AS IT IS DRAFTED, I REALLY DON'T HAVE A LOT TO ADD TO IT, GENERALLY SPEAKING. MAYBE YOU HAVE A SPECIFIC QUESTION THAT, IF ASKED CIVILLY, I WILL TRY TO ANSWER. BUT IF YOU REALLY CARE ABOUT UNDERSTANDING THE PROBLEM, AND YOU CAN'T GET THERE AFTER A GOOD FAITH EFFORT TO WORK THROUGH MY EXPLANATION, I ADVISE YOU TO LOOK UP ONE OF THE NUMEROUS BOOKS ON THE SUBJECT. HOWARD MARGOLIS HAS SEVERAL BOOKS ADDRESSING THIS PROBLEM, BUT THERE ARE NUMEROUS OTHER AUTHORS AS WELL (YOU DON'T NEED TO RELY ON "MY" EXPERT!).]
17% of players have answered correctly.

01-15-2007, 12:44	#1 (permalink)
Tunaltay Sahip :p Giriş: Nov 2006 Konum: Fildişi Kule/Onuncu Köy Mesaj: 10,989 Tecrübe Puanı: 100 Rep Puanı: 13548 Rep Derecesi:	For İntermediate:Edgy Grammar and Reason Introduction: "EST. TIME: 30 MINS PLUS. Human reasoning and language comprehension are closely related capacities. This quiz plays in the area where they most clearly intertwine. No factual knowledge is tested here, 'just' analytic reasoning and language processing." Question 1: If no head injury is too trivial to be neglected, then: if you injure your head, consult a physician. if you don't consult a physician, your head injury must be trivial. don't injure your head unless you consult a physician. don't consult a physician no matter how trivial your head injury may be. Question 2: Only one of the following sentences cannot be disambiguated to parse as a grammatical, meaningful sentence in English, using standard grammatical rules. Which one is the agrammatical sentence? Buffalo buffalo buffalo buffalo. The horse raced past the barn fell jumped. The rat the cat the dog chases hunts hides. The claim the horse he entered was a ringer was false. Question 3: Consider this simple syllogism. '(1) All artists are beekeepers. (2) Some beekeepers are chemists.' If (1) and (2) are both true, then which of the following must be true? Some artists are chemists. None of the other options is correct. Some chemists are artists. Some chemists are not artists. Question 4: Three cards are in a hat. One is red on both sides, one is white on both sides, and one is red on one side and white on the other. I draw a card from the hat, and drop it on the table. The upward-facing side is red. What are the odds that the downward-facing side is also red? Fifty percent. Two out of three. One out of three. None of the other options is correct. Question 5: Tom, Dick and Harry are in prison. One of them has been randomly selected to die in the morning, and the other two will be set free. Their guard knows which one will die, but none of the prisoners does. The guard is under strict instructions not to divulge the identity of the doomed man. Tom is desperate for any information beyond the fact that his odds of death are one in three. He begs the guard to throw him an informational bone. Finally, to shut him up, the guard agrees to reveal only the following: the name of one of Tom's fellow prisoners who will be set free rather than killed. The guard then says that Dick will be set free. After receiving this information from the guard, what is the most accurate calculation Tom can make of the probability that he is the doomed man? None of the other options is correct. One out of two. One out of three. Two out of three. ================================================== ========== ================================================== ========== ================================================== ========== Results for Edgy Grammar and Reason Question 1: If no head injury is too trivial to be neglected, then: The correct answer is don't consult a physician no matter how trivial your head injury may be.. Pragmatic cues are powerful guides to how language should be interpreted, often more powerful than mere niceties of grammar. Actually parsing the sentence yields a meaning contrary to that implied by pragmatic cues or, specifically, here, what I refer to as 'keyword heuristics.' 11% of players have answered correctly. Question 2: Only one of the following sentences cannot be disambiguated to parse as a grammatical, meaningful sentence in English, using standard grammatical rules. Which one is the agrammatical sentence? The correct answer is The horse raced past the barn fell jumped.. 'The horse raced past the barn fell' is a famous garden-path sentence, where 'raced past the barn' is used adjectivally (with respect to 'horse') rather than as a verb phrase. But with 'jumped' at the end it stops making sense. 'Buffalo buffalo buffalo buffalo' implies that bison from a particular city in New York tend to bewilder, deceive or confuse (other) bison. The rat-cat-dog sentence is hard to explain without a whiteboard or a tree-diagram, but try to follow this: (i) 'the cat' is a noun phrase modified by the adjectival phrase 'the dog chases', (ii) 'the cat the dog chases' is the noun phrase which is the subject of the verb 'hunts', whose object is 'the rat', (iii) 'the rat the cat the dog chases hunts' is the noun phrase which is the subject of the verb 'hides', which has no object. The horse-race-ringer-false sentence can be clarified with the following technically superfluous but practically much-needed punctuation, as follows: 'The claim ([that] the horse--[the one] he entered [in the race]--was a ringer) was false.' [Apologies to non-native English speakers, who may have had unwarranted trouble with the buffalo sentence, which is really a vocabulary issue. Nevertheless, the correct answer (the actually agrammatical sentence) cannot be parsed, so you "should" have been able to identify it as the correct answer on pure grounds of grammar, even if you could not eliminate the buffalo sentence as demostrably wrong.] 22% of players have answered correctly. Question 3: Consider this simple syllogism. '(1) All artists are beekeepers. (2) Some beekeepers are chemists.' If (1) and (2) are both true, then which of the following must be true? The correct answer is None of the other options is correct.. [Adding to my explanation for this problem about five years after I first posted it to this site, I note that a (to me) very surprising number of people seem to get this wrong. At time of writing, quiztakers have performed at chance levels on this problem. Although only a handful of people have written to me angrily about this one, unlike problems 4 and 5, which still get me frequent hate-mail, despite all my efforts to ward it off (see below).] It is possible that all of the beekeepers who are chemists are also artists. It is possible that none of the beekeepers who are chemists are also artists. There is no justifiable conclusion about the overlap or non-overlap of chemists and artists. [The paragraph immediately above is my first stab at explaining the problem; let me now expand that explanation a little further.] Let us use the following convention: an 'A' is an artist, a 'B' is a beekeeper, a 'C' is a chemist, an 'AB' is an artist who is also a beekeeper, a 'BC' is a beekeeper who is also a chemist, an 'ABC' is an artist who is also a beekeeper and a chemist, and so on. Here is one set that satisfies both premises of the syllogism: AB, AB, ABC, ABC. In this set, every A is also a B (premise 1), and some Bs are also Cs (premise 2). In addition, some As are also Cs, and all Cs are also As. But here is another set that also satisfies both premises of the syllogism: AB, AB, ABC, ABC, BC, BC. Here, too, every A is also a B (premise 1), and some Bs are also Cs (premise 2). But here, while some As are Cs, as above, this time some Cs are not As. And here is yet another set that also satisfies both premises of the syllogism: AB, AB, B, B, BC, BC. Again, every A is also a B (premise 1), and some Bs are also Cs (premise 2). But in this set, no A is also a C (and no C is also an A). Because each of these possible sets (among many others) satisfies both premises of the syllogism, it is not justified to reach any conclusion about the relationship between Cs and As: some, all or none of the Cs could be As, and some, all, or none of the As could be Cs. I hope this helps! [Writing now more like 6 years on from originally posting this problem. People have started writing to me saying things like, "I JUST KNOW that some of the artists are chemists; I can't prove it, but I JUST KNOW your logic is wrong." PLEASE DO NOT WRITE SUCH A MESSAGE TO ME. In the absence of the additional premise "All beekeepers are also artists" the conclusion "Some artists are chemists" is unjustified. It might be true, but it might not be true. If it might not be true, you CANNOT conclude it "must" be true.] 27% of players have answered correctly. Question 4: Three cards are in a hat. One is red on both sides, one is white on both sides, and one is red on one side and white on the other. I draw a card from the hat, and drop it on the table. The upward-facing side is red. What are the odds that the downward-facing side is also red? The correct answer is Two out of three.. The problem uses three cards (R/R, R/W, W/W), each with two sides. Each time a card is drawn there is thus a 1-in-3 chance that any particular card will be drawn, but from a formal standpoint that is a red herring, because, relevantly, there is a 1-in-6 chance that any particular SIDE will come up. That is, there is a 1-in-3 chance that the R/R card will be drawn, and then, given that R/R has been drawn, there is a fifty-fifty chance that the "A" side will come up and a fifty-fifty chance that the "B" side will come up. (The two sides are functionally identical, but, again, that fact is a red herring from a formal standpoint.) Let us call side "A" of the R/R card R/R:A and side "B" of that card R/R:B. It is helpful to look at this as though we were making a series of observations of the problem, as though it were an experiment. Suppose we "ran" the problem 600 times. (We would expect to see red come up 300 times and white come up 300 times, but that observation would be meaningless from the formal standpoint. Nevertheless, it is probably this fact that triggers many people's incorrect intuitions about the problem.) Relevantly, we would expect the following: R/R to be drawn 200 times (of which we would expect R/R:A 100 times and R/R:B 100 times); R/W to be drawn 200 times (of which we would expect the red side 100 times and the white side 100 times); and W/W to be drawn 200 times (of which we would expect W/W:A 100 times and W/W:B 100 times). So the problem tells us that on this particular instance a red side came up, meaning that we can throw out all observations where a white side came up. We are left with: the 100 times we saw R/R:A, the 100 times we saw R/R:B and the 100 times we saw R/W:A. Of the approximately 300 draws culminating in a red side, approximately 200 of them would have been draws of the R/R card and 100 would have been draws of the R/W card. Thus, if we continued to "run" the "experiment," and every time we saw a red side come up we predicted that we had drawn the R/R card, the prediction would be accurate approximately 2/3 of the time. That is to say, there is a 2/3 probability on any draw that there will be red on the other side of any red draw. * If you still don't understand: before you write to me about this, please, please, try this simple experiment at home. I mean really do this before you write to me. Please. Take 3 quarters, and with a magic marker, write (i) an R on both sides of one of the quarters, (ii) a W on both sides of one, and (iii) an R on one side and a W on the other side of the third quarter. (Technically, the W/W quarter is wholly superfluous, but let's set this up just like the problem in the quiz question. However, if you really see why we don't need to use the W/W quarter, feel free to use just the R/R and R/W quarters.) Also take a piece of paper, and write the numbers from 1 to 30 in a column, with a blank space to the right of each number. Okay, put the quarters in a hat, and pull one out at random with your eyes closed, placing it flat on a table top before you open your eyes. The quiz problem says that red is facing up, so if you get a W, put the quarter back in the hat and repeat. When you have an R facing up, check the other side. Write down what was on the other side (R or W) in the blank corresponding to the number 1 on your piece of paper. Repeat, writing down the result for the downward-facing side in blank number two, and continue repeating (always throwing the coin back and NOT recording the downward-facing side whenever a W is the upward-facing side) until you have all 30 blanks filled. You will have written R approximately 20 times, and W approximately 10 times. * [IF YOU GOT THIS QUESTION WRONG, AND HAVE TROUBLE UNDERSTANDING HOW YOUR ANSWER COULD POSSIBLY BE WRONG, YOU ARE IN GOOD COMPANY: THIS PROBLEM IS WELL STUDIED BY PSYCHOLOGISTS, LOGICIANS, AND GAME THEORISTS, AND IT IS WELL DOCUMENTED THAT EVEN VERY WELL-EDUCATED PEOPLE GET THIS PROBLEM WRONG MORE OFTEN THAN NOT, AND HAVE TROUBLE UNDERSTANDING THE CORRECT ANSWER. IT IS HIGHLY COUNTER-INTUITIVE. IF YOU GOT THE PROBLEM WRONG AND FIND YOURSELF TEMPTED TO CONTACT ME TO TELL ME WHY MY ANSWER IS WRONG AND YOURS IS RIGHT, YOU ARE ONE OF A BIG, BIG CROWD. BUT PLEASE READ ON BEFORE YOU DO SO. FIRST: MY ANSWER IS SIMPLY NOT IN ERROR. THE PROBLEM HAS BEEN EXHAUSTIVELY ANALYZED BY SCHOLARS. IF YOU THINK THAT YOU HAVE UNCOVERED AN ERROR HERETOFORE UNDETECTED BY ANY OF THE MANY ACADEMICS WHO HAVE BUILT CAREERS TRYING TO UNDERSTAND HOW HUMANS REASON OVER PROBLEMS LIKE THIS ONE, THEN I BESEECH YOU, IN THE BOWELS OF CHRIST, CONSIDER THAT YOU MIGHT BE WRONG. SECOND: THE ABOVE IS MY BEST EFFORT IN THIS CONTEXT TO EXPLAIN THE CORRECT ANSWER. I DON'T WANT TO BE CONTACTED BY ANYONE WHO CANNOT HONESTLY SAY THAT THEY HAVE SPENT AT LEAST HALF AN HOUR WORKING THROUGH MY EXPLANATION AND TESTING IT BY SIMULATING THE PROBLEM. DON'T WASTE MY TIME WITH A MESSAGE THAT, AT BEST, FAILS TO TAKE INTO ACCOUNT THE EXPLANATION I HAVE PROVIDED. FINALLY, BECAUSE THE ABOVE IS MY BEST EFFORT TO EXPLAIN THE CORRECT ANSWER TO THE PROBLEM AS IT IS DRAFTED, I REALLY DON'T HAVE A LOT TO ADD TO IT, GENERALLY SPEAKING. MAYBE YOU HAVE A SPECIFIC QUESTION THAT, IF ASKED CIVILLY, I WILL TRY TO ANSWER. BUT IF YOU REALLY CARE ABOUT UNDERSTANDING THE PROBLEM, AND YOU CAN'T GET THERE AFTER A GOOD FAITH EFFORT TO WORK THROUGH MY EXPLANATION, I ADVISE YOU TO LOOK UP ONE OF THE NUMEROUS BOOKS ON THE SUBJECT. HOWARD MARGOLIS HAS SEVERAL BOOKS ADDRESSING THIS PROBLEM, BUT THERE ARE NUMEROUS OTHER AUTHORS AS WELL (YOU DON'T NEED TO RELY ON "MY" EXPERT!).] 18% of players have answered correctly. Question 5: Tom, Dick and Harry are in prison. One of them has been randomly selected to die in the morning, and the other two will be set free. Their guard knows which one will die, but none of the prisoners does. The guard is under strict instructions not to divulge the identity of the doomed man. Tom is desperate for any information beyond the fact that his odds of death are one in three. He begs the guard to throw him an informational bone. Finally, to shut him up, the guard agrees to reveal only the following: the name of one of Tom's fellow prisoners who will be set free rather than killed. The guard then says that Dick will be set free. After receiving this information from the guard, what is the most accurate calculation Tom can make of the probability that he is the doomed man? The correct answer is One out of three.. This problem is tricky for reasons not well understood by psychologists. Some characteristic of human reasoning makes it difficult to grasp the logic, even once it has been laid out openly. I find that providing an analogous illustration helps facilitate people's understanding of this problem. Assume that there are 100 children and 100 boxes. Each box has a different one of the children's names on it. I pick one at random to put a gold star inside; inside all the rest I put a silver star. I then seal all the boxes. Now suppose that Tom is one of the children. Because I selected the box at random, as far as he knows there is a 1% chance that he received the gold star in his box. That is, if we did this whole "experiment" 100 times, in approximately 1 run of the experiment Tom would get the gold star and the other 99 boxes would contain silver stars; and in approximately 99 runs of the experiment Tom would get a silver star, 98 other boxes would contain a silver star, and 1 other box would contain a gold star. Note that in all 100 runs, at least 98 boxes other than Tom's box contain silver stars; that will be true no matter where the gold star is. Of course, _I_ know which box contains the gold star. I can therefore always generate a list of 98 names (always excluding Tom's name) that correspond to boxes containing silver stars. I can do this if Tom's box contains the gold star and I can do this even if Tom's box does not contain the gold star. Suppose that I compile such a list for Tom's benefit, having assured him in advance that the list will contain precisely 98 names of persons other than Tom whose boxes contain silver stars; in other words, that no matter which box contains the gold star the list will reduce the candidate gold-star boxes to two, Tom's and one other. HAVE TOM'S ODDS OF RECEIVING THE GOLD STAR ROCKETED FROM 1% TO 50%? CONSIDER THAT IF WE "RUN" THE "EXPERIMENT" 100 TIMES, WE ONLY EXPECT TOM TO RECEIVE THE GOLD STAR ONCE, AND YET I CAN PRODUCE A LIST OF 98 NON-TOM SILVER-STAR NAMES IN _EVERY SINGLE RUN_. WILL GENERATION OF A 98 NON-TOM SILVER-STAR-NAME LIST CAUSE TOM TO RECEIVE THE GOLD STAR FIFTY PERCENT OF THE TIME? OBVIOUSLY NOT; I CAN GENERATE SUCH A LIST WITHOUT AFFECTING THE ODDS. TOM STILL ONLY HAS A 1% CHANCE OF RECEIVING THE GOLD STAR ON ANY PARTICULAR "RUN." I hope the correspondence to the Prisoner Problem is obvious. Change the number of boxes from 100 to 3, and the gold star to a death warrant, and the logic is identical across the two situations. The information Tom was given had no effect on his own odds. Note that although Tom's odds remain 1 in 3, Harry's odds of being the one to be killed can now be rated at 2 in 3. This is because of the conditions the guard imposed: he said he was generating a one-person non-Tom list of people not being killed; he didn't promise to make a non-Harry list, so the new information that, of persons who are not Tom, Dick is safe doubles the probability that Harry is going down. * Here is ANOTHER way of explaining it. You can set this up and try it at home. In fact, please REALLY DO try this at home before you write to me. Pick three friends to help you, and you be the jailer. Pick one of your friends to be "Tom." Put your friends' names in a hat, and pick one of them at random: that one will be "killed." Tell Tom the name of one of the other two friends who will not be killed. Ask Tom what his odds of being killed are. He will probably say 50%. Reveal to him who will actually be killed. Write down the name of the person to be killed. Do this 30 times. It may be that Tom always thinks he has a 50-50 chance after the candidates for being killed are narrowed down to two people, but in fact, at the end of the day, you will have written Tom's name down only approximately 10 times (1/3 of 30). And of the approximately 15 times that you said "Harry will not be killed" you will have written down "Dick was killed" approximately 10 times (2/3 of 15), and vice versa. All get killed approximately the same number of times over the long haul, but only the calculation of odds for the non-Tom prisoners change as a result of the guard's information. * [IF YOU GOT THIS QUESTION WRONG, AND HAVE TROUBLE UNDERSTANDING HOW YOUR ANSWER COULD POSSIBLY BE WRONG, YOU ARE IN GOOD COMPANY: THIS PROBLEM IS WELL STUDIED BY PSYCHOLOGISTS, LOGICIANS, AND GAME THEORISTS, AND IT IS WELL DOCUMENTED THAT EVEN VERY WELL-EDUCATED PEOPLE GET THIS PROBLEM WRONG MORE OFTEN THAN NOT, AND HAVE TROUBLE UNDERSTANDING THE CORRECT ANSWER. IT IS HIGHLY COUNTER-INTUITIVE. IF YOU GOT THE PROBLEM WRONG AND FIND YOURSELF TEMPTED TO CONTACT ME TO TELL ME WHY MY ANSWER IS WRONG AND YOURS IS RIGHT, YOU ARE ONE OF A BIG, BIG CROWD. BUT PLEASE READ ON BEFORE YOU DO SO. FIRST: MY ANSWER IS SIMPLY NOT IN ERROR. THE PROBLEM HAS BEEN EXHAUSTIVELY ANALYZED BY SCHOLARS. IF YOU THINK THAT YOU HAVE UNCOVERED AN ERROR HERETOFORE UNDETECTED BY ANY OF THE MANY ACADEMICS WHO HAVE BUILT CAREERS TRYING TO UNDERSTAND HOW HUMANS REASON OVER PROBLEMS LIKE THIS ONE, THEN I BESEECH YOU, IN THE BOWELS OF CHRIST, CONSIDER THAT YOU MIGHT BE WRONG. SECOND: THE ABOVE IS MY BEST EFFORT IN THIS CONTEXT TO EXPLAIN THE CORRECT ANSWER. I DON'T WANT TO BE CONTACTED BY ANYONE WHO CANNOT HONESTLY SAY THAT THEY HAVE SPENT AT LEAST HALF AN HOUR WORKING THROUGH MY EXPLANATION AND TESTING IT BY SIMULATING THE PROBLEM. DON'T WASTE MY TIME WITH A MESSAGE THAT, AT BEST, FAILS TO TAKE INTO ACCOUNT THE EXPLANATION I HAVE PROVIDED. FINALLY, BECAUSE THE ABOVE IS MY BEST EFFORT TO EXPLAIN THE CORRECT ANSWER TO THE PROBLEM AS IT IS DRAFTED, I REALLY DON'T HAVE A LOT TO ADD TO IT, GENERALLY SPEAKING. MAYBE YOU HAVE A SPECIFIC QUESTION THAT, IF ASKED CIVILLY, I WILL TRY TO ANSWER. BUT IF YOU REALLY CARE ABOUT UNDERSTANDING THE PROBLEM, AND YOU CAN'T GET THERE AFTER A GOOD FAITH EFFORT TO WORK THROUGH MY EXPLANATION, I ADVISE YOU TO LOOK UP ONE OF THE NUMEROUS BOOKS ON THE SUBJECT. HOWARD MARGOLIS HAS SEVERAL BOOKS ADDRESSING THIS PROBLEM, BUT THERE ARE NUMEROUS OTHER AUTHORS AS WELL (YOU DON'T NEED TO RELY ON "MY" EXPERT!).] 17% of players have answered correctly. __________________ Herşeyi bilirim mi diyor gençlik? Herşeyi yaparım mı diyor ihtiyarlık?

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