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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. # Few mathematical constructs seem as conceptually simple as  Question banks Downloads My Bookmarks Reviews Important topics
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Moderator  Joined: 18 Apr 2015
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Few mathematical constructs seem as conceptually simple as [#permalink]
Expert's post 00:00

Question Stats: 84% (04:01) correct 15% (03:55) wrong based on 13 sessions

Few mathematical constructs seem as conceptually simple as that of randomness. According to the traditional definition, a number is random if it is chosen purely as the result of a probabilistic mechanism such as the roll of a fair die. In their groundbreaking work regarding complexity and the limitations of formal systems, mathematicians Gregory Chaitin and A.N. Kolmogorov force us to consider this last claim more closely.

Consider two possible outcomes of throwing a fair die three times: first, 1, 6, and 2; second 3, 3, and 3. Now let us construct two three-member sets based on the results. Though the first set—{1,6,2}—intuitively seems more random than the second—{3,3,3}, they are each as likely to occur, and thus according to the accepted definition, must be considered equally random. This unwelcome result prompts Chaitin and Kolmogorov to suggest the need for a new standard of randomness, one that relies on the internal coherence of the set as opposed to its origin.
Which of the following best describes the organization of the passage as whole?

A) A concept is introduced; a traditional definition is put forward; a thought experiment is described; a new definition is proposed; the traditional definition is amended as a result.
B) A concept is introduced; a traditional definition is supported by authorities; a thought experiment is described; the implications of the experiment are discussed.
C) A concept is introduced; a traditional definition is considered and rejected; a thought experiment is described; a new definition is proposed.
D) A concept is introduced; a traditional definition is called into question; a thought experiment is described; the implications of the experiment are discussed.
E) A concept is introduced; authorities are called in to reevaluate a definition; a thought experiment is described; the implications of the experiment are considered and rejected.

[Reveal] Spoiler: OA
D

Consider each of the choices separately and select all that apply.

Which of the following is an inference made in the passage above?

❑ The results of the same probabilistic mechanism will each be as likely as the other to occur.

❑ According to the traditional definition of randomness, two numbers should be considered equally random if they result from the same probabilistic mechanism.

❑ Different probabilistic mechanisms are likely to result in similar outcomes.

[Reveal] Spoiler: OA
A and B

_________________
Intern Joined: 28 Mar 2018
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
1st statement in 2nd question,is this not the traditional way.pls explain.
Moderator  Joined: 18 Apr 2015
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Kudos [?]: 1143 , given: 5441

Re: Few mathematical constructs seem as conceptually simple as [#permalink]
Expert's post
I had a hard time to figure it out what you did mean.

Please, could you clarify to me so I can help you?

Regards
_________________
Intern Joined: 28 Mar 2018
Posts: 21
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Kudos [?]: 4 , given: 11

Re: Few mathematical constructs seem as conceptually simple as [#permalink]
in the 2nd question, conflict is between traditional vs modern mathematicians.mathematicians say randomness of an event is contingent on previous event(3,3,3).1st statement is the conflict ,it cannot be inferred from the passage.sorry for my post.
Manager Joined: 26 Jan 2018
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
3rd statement is out of scope of the essay as nothing is mentioned about how different probabilistic mechanisms would give outcomes. Please correct if my reasoning is wrong. Moderator  Joined: 18 Apr 2015
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
1
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Expert's post
Quote:
Though the first set—{1,6,2}—intuitively seems more random than the second—{3,3,3}, they are each as likely to occur, and thus according to the accepted definition, must be considered equally random.

The bold part confirms both answers A and B that actually are formulated like two sides of the same coin.

The third is not mentioned anywhere in the passage.

Hope this helps now.

Regards
_________________ Intern Joined: 18 Sep 2018
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
1
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Choice A is not correct: The results of the same probabilistic mechanism will each be as likely as the other to occur.

It says at the end of Paragraph 1: a number is random if it is chosen purely as the result of a probabilistic mechanism such as the roll of a fair die....mathematicians Gregory Chaitin and A.N. Kolmogorov force us to consider this last claim more closely.

So this inference is NOT made. It is in fact called into question, in the passage.
Intern Joined: 02 Oct 2018
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
njihan wrote:
Choice A is not correct: The results of the same probabilistic mechanism will each be as likely as the other to occur.

It says at the end of Paragraph 1: a number is random if it is chosen purely as the result of a probabilistic mechanism such as the roll of a fair die....mathematicians Gregory Chaitin and A.N. Kolmogorov force us to consider this last claim more closely.

So this inference is NOT made. It is in fact called into question, in the passage.

Choice A is OA.
Also this choice A is about the mathematical feature of probability.
"any result from a probabilistic mechanism is equally likely" is not contradicted anywhere in the passage. It is the basic feature of probability.

The controversy in the passage is on the perception of randomness. We perceive an event where, getting a 1,6,2 on three successive rolls of a dice is random and 3,3,3 as a coincidental event. However the mathematicians contention, is that even 3,3,3 is a random phenomenon and not coincidental. Therefore it is the definition of "random" that is being discussed and not the definition of "probability". Hence Choice A holds true.
Intern Joined: 04 Nov 2018
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
Carcass wrote:

Few mathematical constructs seem as conceptually simple as that of randomness. According to the traditional definition, a number is random if it is chosen purely as the result of a probabilistic mechanism such as the roll of a fair die. In their groundbreaking work regarding complexity and the limitations of formal systems, mathematicians Gregory Chaitin and A.N. Kolmogorov force us to consider this last claim more closely.

Consider two possible outcomes of throwing a fair die three times: first, 1, 6, and 2; second 3, 3, and 3. Now let us construct two three-member sets based on the results. Though the first set—{1,6,2}—intuitively seems more random than the second—{3,3,3}, they are each as likely to occur, and thus according to the accepted definition, must be considered equally random. This unwelcome result prompts Chaitin and Kolmogorov to suggest the need for a new standard of randomness, one that relies on the internal coherence of the set as opposed to its origin.
Which of the following best describes the organization of the passage as whole?

A) A concept is introduced; a traditional definition is put forward; a thought experiment is described; a new definition is proposed; the traditional definition is amended as a result.
B) A concept is introduced; a traditional definition is supported by authorities; a thought experiment is described; the implications of the experiment are discussed.
C) A concept is introduced; a traditional definition is considered and rejected; a thought experiment is described; a new definition is proposed.
D) A concept is introduced; a traditional definition is called into question; a thought experiment is described; the implications of the experiment are discussed.
E) A concept is introduced; authorities are called in to reevaluate a definition; a thought experiment is described; the implications of the experiment are considered and rejected.

[Reveal] Spoiler: OA
D

Consider each of the choices separately and select all that apply.

Which of the following is an inference made in the passage above?

❑ The results of the same probabilistic mechanism will each be as likely as the other to occur.

❑ According to the traditional definition of randomness, two numbers should be considered equally random if they result from the same probabilistic mechanism.

❑ Different probabilistic mechanisms are likely to result in similar outcomes.

[Reveal] Spoiler: OA
A and B

Could someone please explain why choice D is correct for the first question. I chose C but would love to know the reasoning behind D. Manager Joined: 27 Feb 2017
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Re: Few mathematical constructs seem as conceptually simple as [#permalink]
1
KUDOS
Mercychee wrote:
Carcass wrote:

Few mathematical constructs seem as conceptually simple as that of randomness. According to the traditional definition, a number is random if it is chosen purely as the result of a probabilistic mechanism such as the roll of a fair die. In their groundbreaking work regarding complexity and the limitations of formal systems, mathematicians Gregory Chaitin and A.N. Kolmogorov force us to consider this last claim more closely.

Consider two possible outcomes of throwing a fair die three times: first, 1, 6, and 2; second 3, 3, and 3. Now let us construct two three-member sets based on the results. Though the first set—{1,6,2}—intuitively seems more random than the second—{3,3,3}, they are each as likely to occur, and thus according to the accepted definition, must be considered equally random. This unwelcome result prompts Chaitin and Kolmogorov to suggest the need for a new standard of randomness, one that relies on the internal coherence of the set as opposed to its origin.
Which of the following best describes the organization of the passage as whole?

A) A concept is introduced; a traditional definition is put forward; a thought experiment is described; a new definition is proposed; the traditional definition is amended as a result.
B) A concept is introduced; a traditional definition is supported by authorities; a thought experiment is described; the implications of the experiment are discussed.
C) A concept is introduced; a traditional definition is considered and rejected; a thought experiment is described; a new definition is proposed.
D) A concept is introduced; a traditional definition is called into question; a thought experiment is described; the implications of the experiment are discussed.
E) A concept is introduced; authorities are called in to reevaluate a definition; a thought experiment is described; the implications of the experiment are considered and rejected.

[Reveal] Spoiler: OA
D

Consider each of the choices separately and select all that apply.

Which of the following is an inference made in the passage above?

❑ The results of the same probabilistic mechanism will each be as likely as the other to occur.

❑ According to the traditional definition of randomness, two numbers should be considered equally random if they result from the same probabilistic mechanism.

❑ Different probabilistic mechanisms are likely to result in similar outcomes.

[Reveal] Spoiler: OA
A and B

Could someone please explain why choice D is correct for the first question. I chose C but would love to know the reasoning behind D.

Q. 1. Process of Elimination works best here. We can instantly eliminate B C and E because traditional definition is not supported or rejected or re-evaluated. If you look at the second sentence, they just state the traditional definition and in third sentence, it says to re-consider it. As for A and E, A cannot be the answer, simply because tradiitonal definition is not really chnaged. Two mathematicians do propose it should be chnaged but that hasn't taken place yet. So E is the answer and once you really put these pieces together, it does make sense. Their rethinking the definition is an implication of their experiment. Re: Few mathematical constructs seem as conceptually simple as   [#permalink] 03 Dec 2018, 15:00
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