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Few mathematical constructs seem as conceptually simple as

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Few mathematical constructs seem as conceptually simple as [#permalink] New post 03 Jun 2017, 01:18
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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


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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 11 Apr 2018, 14:16
1st statement in 2nd question,is this not the traditional way.pls explain.
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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 12 Apr 2018, 06:04
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I had a hard time to figure it out what you did mean.

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

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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 12 Apr 2018, 07:11
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.
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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 12 Apr 2018, 07:24
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.
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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 12 Apr 2018, 07:28
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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.

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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 30 Oct 2018, 09:15
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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.
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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 09 Nov 2018, 10:56
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.
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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 10 Nov 2018, 17:54
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.
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Re: Few mathematical constructs seem as conceptually simple as [#permalink] New post 03 Dec 2018, 15:00
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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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