Are these two expressions “equal”? - Gambling and Probability

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Are these two expressions “equal”?

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12-02-2017 , 08:18 PM

robert_utk

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Short and sweet.

Is this an allowed description of a uniform real distribution of numbers?

From a formal mathematics perspective, can I say that:

The random real numbers (0,1)

Equals....

The random real numbers [1-.999999etc,.9999etc]

Does forcing one of the inclusive boundaries to be an expression of the other inclusive boundary break a formal rule?

Thanks!

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12-02-2017 , 08:30 PM

RustyBrooks

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Well, the problem is, I think, that .9 repeating equals 1. It's not "close to 1", it is 1. So you turned (0, 1) into [0, 1] and those are not the same.

ETA: really I keep going back and forth. I could see an argument either way.

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12-02-2017 , 08:42 PM

robert_utk

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Would both be an acceptable way to say “any number between 0 and 1” ? Granted, the simpler expression should be used for simplicity sake, but can you define one side as an expression of the other, such as (1-1,1)?

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12-02-2017 , 08:50 PM

RustyBrooks

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Sure? Theoretical math is not real big on doing arithmetic so it's common to see ranges like (1/2, sqrt(2)) and no one is going to make you say (0.5, 1.414...)

But you should prefer to just say (0, 1) imo because this is literally what it's meant for

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12-02-2017 , 08:56 PM

robert_utk

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Is (1-x,x) a valid expression of real numbers, so long as x is real?

Thank you RustyBrooks

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12-02-2017 , 09:03 PM

RustyBrooks

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Quote:

Originally Posted by robert_utk

Is (1-x,x) a valid expression of real numbers, so long as x is real?

I dunno, seems fine to me. The rules are pretty fluid as long as it's clear what you mean.

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12-02-2017 , 09:06 PM

robert_utk

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Yay!

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12-02-2017 , 10:27 PM

heehaww

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Pretty sure that in order to say (1-x, x) you have to specify that x >= 1/2 (if x=1/2 then that's the null set so that works).

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12-03-2017 , 12:52 PM

Carpal \'Tunnel

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Quote:

Originally Posted by RustyBrooks

There is not a (correct) argument either way. 0.9999 repeating is a real number and 1 is a real number. If 0.99999 repeating is less than 1 there exists a rational number x such that 0.99999 repeating < x < 1. No such rational number exists, so 0.9999 repeating equals 1, by the density of rational numbers.

If you were not saying you go back and forth on if they are equal, I apologize for misreading what you meant.

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12-03-2017 , 10:54 PM

#10

RustyBrooks

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No, they are definitely equal.

But consider, what's the largest number in the range (0, 1)? If you were going to rewrite the expression (0, 1) as [0, x], what's x? I can't think of a better candidate than .9 repeating.

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12-03-2017 , 11:26 PM

#11

heehaww

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There is no largest number in the range, so you can't rewrite it in square brackets except as the expression
[0,1] \ [0,0]U[1,1]

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12-04-2017 , 03:08 AM

#12

nickthegeek

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Join Date: Sep 2011 Posts: 273

Quote:

Originally Posted by RustyBrooks

There is no "largest number" in that set. A set doesn't have to have a maximum. For instance, what's the largest number in R?

The (0,1) range does have an infimum and a supremum.

You can't find an x such as (0,1) and (0,x] are the same set. If you could, there wouldn't any need of discriminate between "(" and "[" in the interval notation.

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12-04-2017 , 10:35 AM

#13

robert_utk

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Quote:

Originally Posted by nickthegeek

I knew you guys and gals would know the answer, and help me learn as well! I know I should crack a book and learn such things on my own. However, my time is limited and devoted to a different problem.

What I *should* have asked was:

Can I say that....

(0,1) is interchangeable with (1-1,1)

Which I think is yes, from the help here from RustyBrooks

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12-04-2017 , 04:14 PM

#14

nickthegeek

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Join Date: Sep 2011 Posts: 273

Quote:

Originally Posted by robert_utk

Can I say that....

(0,1) is interchangeable with (1-1,1)

I can hardly see the point of this, but you can of course, since 1-1=0. (0,1) is even interchangeable with (44-44,73-72), but I can't see how this "result" can be of any use.

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12-06-2017 , 02:54 PM

#15

robert_utk

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When you want to relate the results of a problem involving (0,1) to the theoretical results of the same problem in (0,x) where x>1 then a particular nomenclature will arise. Thus the bounds become an expression of each other that are no as simple as just saying (0,x+1) if the new bounds are in fact (x-1,2x) or somesuch

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