THE ULTIMATE RIDDLE PART TWO.

dukerdog wrote:

I understand the concept of infinity enough to understand that you can make definitive statements about it. 0.999 followed by an infinite number of 9's IS EQUAL to 1. It doesn't approach 1, it is equal to 1. If an infinite number of prisoners are picked, the probability that they will all be picked doesn't approach 1, it is 1.

By using the word "eventually" in the problem statement, againtocarthage implied that the time frame was infinite. Qualifying the answer with statements about finite time periods is as irrelevant as broken light bulbs and taking dumps in the corner. None of those things are part of the

Read my post above, which addresses this. Sure, the problem says "eventually," but it also requires the prisoners to make their declaration, which creates a finite end date for this process (step 3 above).

I'd like to reiterate that I'm not trying to criticize the answer proposed. We've got it right, and quibbling about this won't improve the answer at all. This is just an interesing side note that people don't seem to get.

math teacher wrote:

4. Then the probability that any single given prisoner has already been in the room on Day N is (1-1/100)^N.* This is a series of nines; but it is a FINITE series of nines. Thus it is not yet equal to one.

I think you mean 1-(1-1/100)^N. Otherwise I think you've got a good explanation.

u r wrong wrote:

In practical terms you my be correct, but theoretically there does exist a miniscule chance that Bob could be picked forever. This one does approach zero, but would never equal zero.

Any number with an infinite nubmer of nines to the right of the decimal point is EXACTLY equal to an integer. There is NOTHING missing. It's basic mathematics that anyone who passed Algebra 1 should understand.

Until you guys understand that these two things are exactly the same thing, there is no hope of convincing you.

0.999...=1

Eventually, all prisoners will be picked.

Saying that at some point in the future you have to stop adding 9's or stop picking prisoners in order to somehow make the problem "practical" doesn't make the above two statements any less true.

dukerdog wrote:

0.999...=1

True.

dukerdog wrote:

Eventually, all prisoners will be picked.

Also true, over an infinite extent of time.

dukerdog wrote:

Saying that at some point in the future you have to stop adding 9's or stop picking prisoners in order to somehow make the problem "practical" doesn't make the above two statements any less true.

You're right, it doesn't make those statements any less true. But it makes them irrelevant.

To turn around your own argument, saying that "eventually every prisoner will be picked" doesn't make it any less true that the problem requires an end date (even as it uses the word eventually), and on any given end date there is still a chance that every prisoner hasn't been picked.

It's a bit of a paradox: it is guaranteed that "eventually" every prisoner will be picked, but on any day we pick there remains a chance that we haven't met this condition; so there is no day on which is guaranteed that every prisoner has made it into the room, only a continued faith that it will "eventually" happen--which is useless for the prisoners' purpose. This is the jump you're still missing. Go back to where I broke it down to steps and see if you can find any break in the logic.

I think one other problem is that people are arguing different things here.

Your argument = eventually all prisoners will be picked.

I agree with this.

My argument = there does not exist a day on which the probability that all prisoners has been picked is 100%.

math teacher wrote:

You're right, it doesn't make those statements any less true. But it makes them irrelevant.

The problem asked a question about infinity. The only answer you are required to give is about infinity.

Picking an end date is answering a question about a finite length of time, which the problem did not ask you to answer.

This is the jump you're still missing. Go back to where I broke it down to steps and see if you can find any break in the logic.

The breakdown in all your logic is picking an end date. The problem didn't ask you to do that, and you certainly don't need to do that to answer a question about infinity.

I hereby declare that I'm going to stop with this. Unless, of course, you say something outrageous. ;-)

dukerdog wrote:

The problem asked a question about infinity. The only answer you are required to give is about infinity.

Picking an end date is answering a question about a finite length of time, which the problem did not ask you to answer.

dukerdog wrote:

The breakdown in all your logic is picking an end date. The problem didn't ask you to do that, and you certainly don't need to do that to answer a question about infinity.

I guess I won't expect a response to this, but I'm claiming that the question is not about infinity:

againtocarthage wrote:

What is the optimal plan they can agree on, so that eventually, someone will make a correct assertion?

In order for someone to make a correct assertion, there must be an end date. If there is no end date, there is no correct assertion. The plan itself requires an end date to be implemented.

.9999999...... does NOT =1

you people are idiots.

.999999..... = .99999.....

or: limit of [1-(1/x)] as x approaches infinity=1

or in words, it approaches but never equals 1.

you cannot equate an infinite CONCEPT, that's what infinity is afterall, with a real number.

as it relates to this problem, over an infinite time period the probablity equals 1, but in no way does that allow you to say .99999....=1

butters wrote:

.9999999...... does NOT =1

you people are idiots.

.999999..... = .99999.....

or: limit of [1-(1/x)] as x approaches infinity=1

or in words, it approaches but never equals 1.

you cannot equate an infinite CONCEPT, that's what infinity is afterall, with a real number.

Jeez, here we go again. Good thing I live in South Dakota and have little else to do.

It's funny that you exclude the concept of infinity from real numbers. Of course infinity is not a real number, but the concept itself is required for our understanding of real numbers. One way of defining real numbers, in fact, is any number that can be represented by an infinite decimal expansion (there's that word!). Under this defintion, we also must state that 1 = 0.999...

Since 0.999... and 1 are real numbers, were they not equal, we would have to find some number between the two. What would that number be?

Jeez, here we go again. Good thing I live in South Dakota and have little else to do.

It's funny that you exclude the concept of infinity from real numbers. Of course infinity is not a real number, but the concept itself is required for our understanding of real numbers. One way of defining real numbers, in fact, is any number that can be represented by an infinite decimal expansion (there's that word!). Under this defintion, we also must state that 1 = 0.999...

Since 0.999... and 1 are real numbers, were they not equal, we would have to find some number between the two. What would that number be?

i'll say it again, you are an idiot. my only mistake was in using the term "real" when i meant rational. you cannot equate a nonrational number with a rational number.

butters wrote:

i'll say it again, you are an idiot. my only mistake was in using the term "real" when i meant rational. you cannot equate a nonrational number with a rational number.

Rational numbers are a subset of the real numbers. And in fact 0.999... is rational as it is equal to 1.

Similarly: is 0.333.. equal to 1/3? Yes.

This is a quote from Wikipedia, but yes, it is accurate:

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like {3, 3.1, 3.14, 3.141, 3.1415,…} converges to a unique real number.

So let's take the sequence {.9, .99, .999, .999, ...}. Obviously this sequence converges to the unique real number 0.9999...

I'm not gonna get all into Cauchy convergence here, mostly because I don't remember it, but if you can understand how the limit of (1-1/100)^n as n goes to infinity is 1, you have a grasp of fhow this sequence above also converges to 1. Because the sequence must converge to a UNIQUE real number, this means 1 = 0.999...

Or, if you want me to just throw a quote at you and cite my sources, this is from "Real Analysis with Real Applications," Prentice Hall 2002, Davidson and Donsig, p. 35:

"[S]ome real numbers (precisely all those numbers with a finite decimal expansion) have two different expansions, one ending in an infinite string of zeroes, and the other ending with an infinite string of nines. For example, 0.125 = 0.12499999...."

math teacher wrote:

but if you can understand how the limit of (1-1/100)^n as n goes to infinity is 1

Look. I told you this once earlier, but this is the second time you have made this mistake. (1-1/100)^n DOES NOT CONVERGE TO ONE. That is .99^n. It gets smaller every time. Now your argument otherwise is fine and should be known by anyone who has taken an intro real analysis course, but I am just a little annoyed I mentioned the mistake above but you ignored it.

For those of you who insist that .9999999... is not 1, well you are wrong. Unfortunately a proof, while not particularly difficult, is probably above most of your levels.

kartelite wrote:

Look. I told you this once earlier, but this is the second time you have made this mistake. (1-1/100)^n DOES NOT CONVERGE TO ONE. That is .99^n. It gets smaller every time. Now your argument otherwise is fine and should be known by anyone who has taken an intro real analysis course, but I am just a little annoyed I mentioned the mistake above but you ignored it.

Haha, whoops. Yeah, didn't really think that one through, tahnks.