Mathematical principles of probability entail that for any future

Question

Mathematical principles of probability entail that for any future event, the probability that it will occur is at least as great as the probability that both it and some other given event will occur. Consider, for example, the following statements that were shown to subjects in a 1998 study.

X. The percentage of adolescent smokers in Texas will decrease at least 15% from current levels by September 1,1999.

Y. The cigarette tax in Texas will increase by $1.00 per pack in 1999.

Z. The cigarette tax in Texas will increase by $1.00 per pack in 1999 and the percentage of adolescent smokers in Texas will decrease at least 15% from current levels by September 1,1999.

Z ("Y and X") could not have been more probable than X. Nevertheless, many of the subjects judged Z to be more probable than X. This mistaken form of reasoning, displayed with surprising frequency in various studies in addition to the 1998 study, is known as the "conjunction fallacy".

A number of researchers have offered alternative explanations for the seeming manifestations of the mistake, thus arguing that the fallacy is less widely committed than the various studies would indicate. Some have claimed that research subjects can take "probability" in a sense that does not conform to the mathematical principles of probability. Detailed descriptions of some such conceptions of "probability" have been developed under the names of "confirmation" and "support." Other researchers would claim, correctly, that subjects shown Z("Y and X") and X simultaneously will sometimes think of X as involving the negation of Y - as a claim that the percentage of adolescent smokers in Texas will decrease, but without the $1.00 increase in the cigarette tax.

However, although the subjects in the 1998 study were to consider X and Z simultaneously, the statements were presented in terms of bets rather than explicit requests for judgments of relative probability. Subjects were asked to choose between Z and X, with a chance of winning $50.00 if the chosen statement turned out to be true. Terms such as "most probable", "likely", etc., were thus avoided and  the interpretation of X in conjunction with the negation of Y was thereby eliminated.  And with these alternative explanations eliminated, many of the subjects nonetheless bet on Z rather than X.   1 
 1. The passage most strongly indicates that the author would agree with which of the following statements?

A. None of the subjects in the various studies other than the 1998 study who seemed to commit the conjunction fallacy actually did commit it.
B. People who have studied the mathematical principles of probability are very unlikely to commit the conjunction fallacy.
C. The conjunction fallacy is rarely committed outside of betting contexts.
D. Many of the subjects in the various studies in addition to the 1998 study probably committed the conjunction fallacy.
E. The conceptions of "probability" that underlie everyday use of the word rarely, if ever, conform to the mathematical principles of probability.

2 
 2. If the claims of the passage are correct, then which of the following best explains why the interpretation of X in conjunction with the negation of Y was eliminated as claimed from the highlighted text?

A. Most of the subjects of the 1998 study recognized that cigarette taxes tend to decrease adolescent smoking.
B. Most of the subjects of the 1998 study interpreted X so as to also include the additional information of Y (as "X and Y " or "Z").
C. Subjects of the 1998 study preferred winning $50 to winning less amount.
D. Subjects of the 1998 study who bet on X would win $50 only if Y turned out to be true.
E. Subjects of the 1998 study who bet on X could have won the $50 whether or not Y turned out to be false.

3 
 3. The final sentence of the passage mentions the elimination of "alternative explanations" of the 1998 study results. The passage most strongly suggests that the author wanted to eliminate the explanations in order to

A. support the claim that the concept of probability is usually interpreted so as not to conform to the mathematical principles of probability
B. counter the claim that the conjunction fallacy is in fact a mistake of reason
C. support an explanation of why some people commit the conjunction fallacy even when they bet on particular outcomes without making explicit probability judgments
D. counter the claim of some researchers that the conjunction fallacy is less frequently committed than various studies would seem to indicate
E. support the point that the conjunction fallacy was not committed by the subjects in studies prior to the 1998 study

4 
 4. The use of bets in the 1998 study was intended to deflect objections that would be based on which of the following issues?

A. The possibility of research subjects interpreting "probability" so as not to conform to the mathematical principles, and their interpretation of X to include additional information
B. The possibility of research subjects interpreting "probability" so as not to conform to the mathematical principles, and the lack of motivation of some of the subjects
C. Failure of research subjects to recognize that adolescent smoking could decrease even when the cigarette tax remains the same
D. The interpretation of X by some study subjects to include additional information and their lack of concentration on the assigned tasks
E. The fact that some of the research subjects did not commit the conjunction fallacy

Blackcrow1972 · Accepted Answer

This is surely a 805+ level passage. 
So don't get discouraged if you even get all of the questions wrong.

Sajjad1994 · Answer

QUESTION 2:

If subjects who bet on X could win $50 regardless of whether Y turned out to be false, then the interpretation of X in conjunction with the negation of Y would not impact the potential winnings. In other words, the bet on X is independent of the truth or falsity of Y.

This would indeed eliminate the need for subjects to consider the negation of Y when placing their bets, aligning with the passage's statement that the interpretation of X in conjunction with the negation of Y was eliminated in the betting scenario.

Answer: E

sayan640 · Answer

2. If the claims of the passage are correct, then which of the following best explains why the interpretation of X in conjunction with the negation of Y was eliminated as claimed from the highlighted text?

a) Most of the subjects of the 1998 study recognized that cigarette taxes tend to decrease adolescent smoking.
b) Most of the subjects of the 1998 study interpreted X so as to also include the additional information of Y (as "X and Y " or "Z").
c) Subjects of the 1998 study preferred winning $50 to winning less amount.
d) Subjects of the 1998 study who bet on X would win $50 only if Y turned out to be true.
e) Subjects of the 1998 study who bet on X could have won the $50 whether or not Y turned out to be false.

Notice this part ,
 Other researchers would claim, correctly, that subjects shown Z("Y and X") and X simultaneously will sometimes think of X as involving the negation of Y - as a claim that the percentage of adolescent smokers in Texas will decrease, but without the $1.00 increase in the cigarette tax.

That X can be a function of Y and can depend on Y's outcome. This possibility was ruled out. X stopped being dependent on Y.
The interpretation of X in conjunction with the negation of Y was eliminated so that the Subjects of the 1998 study who bet on X could have won the $50 whether or not Y turned out to be false. Still many of them went to select Z.

E is the answer.  MartyMurray   GMATNinja   KarishmaB

GMATNinja · Answer

QUESTION 2:

This sounds good! Something about the $50 bet eliminated the "interpretation of X in conjunction with the negation of Y". Choice (E) adequately explains why that would be.

And why did the author want to eliminate that interpretation (moving on to Question 3)? Because it would lead to an alternative explanation for the study results: maybe the subjects DID correctly understand that Z can't be more probably than X, but they picked Z because they thought picking "X" actually meant picking "X will occur and Y will not".

By eliminating that and other alternative explanations (e.g. the stuff about taking "probability" in a sense that does not conform to the mathematical principles of probability), the author strengthens support for the "conjunction fallacy" (i.e. that the study subjects exhibited a mistaken form of reasoning).

In other words, the author shoots down the alternative explanations in order to "counter the claim of some researchers that the conjunction fallacy is less frequently committed than various studies would seem to indicate". So choice (D) is correct for question 3.

I hope that helps!

Blackcrow1972 · Answer

Q2
Key Difference Between B and E:
B suggests that subjects might have still been confused and interpreted X as implicitly including or excluding Y, which the setup of the experiment explicitly avoided.
E explains that the betting framework clarified X’s independence from Y, thus removing the possibility of such confusion.

Blackcrow1972 · Answer

Q3
The author describes the elimination of "alternative explanations" to address claims made by some researchers that the conjunction fallacy is less widespread than studies suggest. By framing the task as a bet and removing potential sources of confusion (e.g., interpreting X as "X without Y"), the study isolates the conjunction fallacy itself. Even with these alternative explanations eliminated, subjects still committed the fallacy, supporting the argument that the conjunction fallacy is indeed a frequent and genuine mistake.

purplelemonsoda · Answer

Q4: (A):
"The possibility of research subjects interpreting 'probability' so as not to conform to the mathematical principles, and their interpretation of X to include additional information."

This matches the two main issues addressed by the use of bets:
Subjects might interpret "probability" in non-mathematical ways (like "confirmation").
Subjects might interpret X as excluding Y, introducing additional information.
✔️ Correct Answer: This is the most accurate description of the purpose of the bets.

(B):
"The possibility of research subjects interpreting 'probability' so as not to conform to the mathematical principles, and the lack of motivation of some of the subjects."

The passage does not mention lack of motivation as a concern or reason for using bets.
Motivation is irrelevant to explaining the conjunction fallacy in this context.
❌ Incorrect: "Lack of motivation" is not a factor.

(C):
"Failure of research subjects to recognize that adolescent smoking could decrease even when the cigarette tax remains the same."

This is related to one of the fallacy's misinterpretations (X being taken as excluding Y), but it does not cover the full scope of issues addressed by the bets.
The bets were also used to address non-mathematical interpretations of "probability," which is not captured here.
❌ Incorrect: This is too narrow and misses a major point.

(D):
"The interpretation of X by some study subjects to include additional information and their lack of concentration on the assigned tasks."

The bets did address the issue of interpreting X to include additional information.
However, lack of concentration is not mentioned in the passage as a concern.
❌ Incorrect: "Lack of concentration" is unsupported.

(E):
"The fact that some of the research subjects did not commit the conjunction fallacy."

The passage emphasizes that many subjects still committed the conjunction fallacy, even with bets.
The purpose of the bets was not to explain why some subjects avoided the fallacy, but to eliminate alternative explanations for why others committed it.
❌ Incorrect: This misrepresents the purpose of the bets.

bb · Answer

Question 1:   The passage most strongly indicates that the author would agree with which of the following statements?

A) Outside of the scope; author never claims none of the earlier subjects committed the fallacy.

B) Irrelevant. Study of math isn’t discussed; no support.

C) Too narrow of a scope since passage never limits the fallacy to betting situations.

D) Correct. In the passage, even after removing other plausible reasons for choosing ZZZ (“Y and X”) over XXX in the 1998 setup, “many of the subjects nonetheless bet on Z rather than X.” That shows the author believes the conjunction fallacy really was committed by a large group of folks in other studies.

E) The author rebuts this theory and says it did not play a role, so this statement is contradictory

Question 3. The final sentence of the passage mentions the elimination of "alternative explanations" of the 1998 study results. The passage most strongly suggests that the author wanted to eliminate the explanations in order to

A) This one is tempting as the passage talks about it but this changes the intent of the author. Very tricky. However, this is not what the author was trying to do (this narrows the scope quite a bit). Instead of trying to prove the probability definition, the author tried to eliminate it as a distraction.

B) I don't know that it is the purpose either. This is not discussed really. Just random answer choice.

C) Also tempting. This is true but again not the author's intention for this sentence. But the author was not trying to explain here WHY some people make the mistake; but rather wanting to clear out arguments against his finding.

D) This seems to be the purpose. By designing the 1998 experiment to remove the confusion related to probability definition and the “X means not-Y” misinterpretation, the author can show that high rates of choosing Z are actually real and higher than critics were willing to accept.

E) This is a silly one, easy to eliminate.

Robo_123 · Answer

Looks like a tough passage. How could one save time and approach this or any passage that is difficult to comprehend?

miag · Answer

Hi,

There are lots of RC strategies by various prep companies, and also a compiled article by gmatclub, sharing these below:

Hope this is useful!

Robo_123

Looks like a tough passage. How could one save time and approach this or any passage that is difficult to comprehend?

gullyboy09 · Answer

Hi  MartyMurray ,  KarishmaB  can you please help me with second question, perhaps I am not getting it completely. My issue with E is that even if they could have won 50$ whether or not Y turned out to be false, but were they aware of it? I mean they can still be choosing Z over X thinking X includes the negation of Y, whereas in reality like E mentioned that's not a case. (But its not mentioned that subjects knew about it that they will win 50 irrespective of Y). OR maybe I am not able to understand the question correctly. Please help.

2. If the claims of the passage are correct, then which of the following best explains why the interpretation of X in conjunction with the negation of Y was eliminated as claimed from the highlighted text?

e) Subjects of the 1998 study who bet on X could have won the $50 whether or not Y turned out to be false.

guddo · Answer

2. If the claims of the passage are correct, then which of the following best explains why the interpretation of X in conjunction with the negation of Y was eliminated as claimed from the highlighted text?

The passage says that in ordinary probability questions, subjects might wrongly read X as “X but not Y.” But in the 1998 study, the subjects were choosing bets.  The key point  is that a bet on X would win whenever X was true,  whether Y was true or false . So X was not naturally interpreted as “X and not Y.”

(A) Most of the subjects of the 1998 study recognized that cigarette taxes tend to decrease adolescent smoking.

This does not explain why the “X and not Y” interpretation was eliminated. It is about what subjects believed about the relationship between taxes and smoking.

(B) Most of the subjects of the 1998 study interpreted X so as to also include the additional information of Y, as “X and Y” or “Z.”

This is wrong. The passage says the study eliminated the interpretation of X as “X and not Y,” not that subjects interpreted X as Z.

(C) Subjects of the 1998 study preferred winning $50 to winning less amount.

This is irrelevant. The issue is not whether subjects wanted to win money, but what event would make the bet on X pay off.

(D) Subjects of the 1998 study who bet on X would win $50 only if Y turned out to be true.

This is false. A bet on X would win if X turned out to be true, regardless of whether Y was true.

(E) Subjects of the 1998 study who bet on X could have won the $50 whether or not Y turned out to be false.

This is correct. If X happened, the subject who chose X would win whether Y also happened or did not happen. So X was not being treated as “X and not Y”; it was simply X.

Answer: (E)

egmat · Answer

Hi gullyboy09,

Your reading of E is right; the piece to let go of is the "were they aware of it?" part.

What the question is actually asking

The "X as X-and-not-Y" reading is an alternative explanation that some researchers offered to argue the subjects weren't really committing the fallacy: maybe they read X as "smoking drops AND the tax does not rise." Question  2  asks why the study's  design  removes that explanation. So the right answer should be a feature of the setup, not a claim about what subjects consciously believed.

Why the bet eliminates the reading

Notice where that reading came from: it shows up when you ask which is "more probable," X or Z. Because Z is the "X and Y" case, X by contrast can get read as "X but not Y." The 1998 study dropped probability language and used a bet instead - win  $50  if your chosen statement turns out true. Now a bet on X simply wins whenever X is true, no matter what Y does. The not-Y part changes nothing about the payoff, so there is no longer any reason to smuggle it into X. That independence is exactly what E states.

On awareness

The payout rule is true whether or not a subject stops to notice it - and the rule itself is what kills the negation reading. The researchers built X's independence from Y into the design; they did not need subjects to reason it out. You are adding a requirement the question never asks for.

A simpler version to feel it

I offer you: "win  $50  if it rains tomorrow." You win if it rains, whether or not the temperature also climbs. You would never read "it rains" as "it rains and the temperature stays flat" - the bet ignores temperature entirely, and that holds even if you never think about it.

Answer: E

Mathematical principles of probability entail that for any future

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Mathematical principles of probability entail that for any future

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