April 5th, 2017, 03:59 PM  #41  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
Quote:
I plead ignorance, being a citizen of the colonies. I have my musket ready in case any redcoats show up. But if it happened in the past, the odds are either 1 or 0 depending on whether it happened or not. You are now discussing the philosophy of probability. Is probability inherent in the event itself? Or only a measurement of our ignorance? If I flip a coin and it lands on the ground, what are the odds it's heads before I look at it? What are the odds for you if you look at it? If the two numbers are different (1/2 for me, certainty one way or the other for you) then odds reflect the state of our knowledge and say nothing about events themselves. I must say I don't see the relevance of this line of thought to the subject. But I don't know much about Collatz or probabilistic proofs. Last edited by Maschke; April 5th, 2017 at 04:08 PM.  
April 5th, 2017, 04:45 PM  #42  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,148 Thanks: 2386 Math Focus: Mainly analysis and algebra  Quote:
You explicitly stated that he's making an error. There's nothing in his view of the problem that stops it being Collatz, is there? You both have the same operations in the same order, just grouped differently. Clearly the odds I was talking about were the prior odds in both cases. Nothing we know about the universe makes higher life forms remotely likely.  
April 5th, 2017, 09:40 PM  #43 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory 
Maschke, I don't think invoking (3n+1)/2 is confusing in any way. It is just a different way of stating the problem other than just phrasing 3n+1 yet we know very well the next number would be even. what we don't know is the outcome after we divide by 2. That's where probability plays its role because the number could either be odd or even. That's what we are sure about. And the formula is what would determine it.

April 5th, 2017, 10:20 PM  #44  
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory  Quote:
On the other point, indeed if we ever flip a coin a million times, indeed there is a chance, however slim, that we would have all heads. But what needs to be proven is that if we flip the coin an infinite number of times, the probability of getting all heads is zero. That can be proven.  
April 5th, 2017, 10:28 PM  #45  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
If we perform the thought experiment of flipping infinitely many fair coins, one for each natural number, the probability that the first one is heads is $\frac{1}{2}$; the probability the first two are heads is $\frac{1}{4}$; and the probability that the first $n$ are heads is $\frac{1}{2^n}$. As $n$ increases without bound, the probability that the first $n$ are heads goes to zero. This is true. It is nevertheless the case that it is conceivable that infinitely many coins may land on heads. Each coin flip is independent of all the others. The coins don't "know" what the other coins are doing. So even though the probability is zero, it could happen. It's just incredibly unlikely. Another thought experiment is to "throw a dart at the real number line." The probability of hitting a rational is zero; but there are rationals on the line, lots of them. You might get lucky and hit one. In fact if you flip infinitely many coins, the odds of any particular sequence is zero. Yet some sequence must occur. If you throw a dart at the real line, you must hit SOME real number, but the odds of you hitting that particular number are zero. This is basic infinitary probability theory. Here is yet another striking example. We can define the asymptotic density of a set of natural numbers as the limit of the percentage of members of that set in the first $n$ numbers, as $n$ goes to infinity. So for example the asymptotic density of the even numbers is $\frac{1}{2}$ just as we'd expect. It turns out that the asymptotic density of the primes is zero. In other words if you pick a random natural number, it's not prime. But there are plenty of primes. [Note  asymptotic density is not a probability measure for technical reasons, so "picking a natural number at random" must be taken with a grain of salt. There is no uniform probability measure on the natural numbers]. http://math.stackexchange.com/questi...ofprimeszero tl;dr: Even if you prove that the probability that Collatz fails is zero, it still might fail. Last edited by Maschke; April 5th, 2017 at 10:50 PM.  
April 5th, 2017, 10:40 PM  #46  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
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I disclaim that if I'm wrong I'm wrong, this [the philosophy of probability] is not an area I have expertise in. Someone will need to show me I'm wrong though. Last edited by Maschke; April 5th, 2017 at 10:48 PM.  
April 5th, 2017, 11:45 PM  #47 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory 
I think we have finally made it. We have the full proof. But still, there is a little more studies to be done on the hailstone sequence, stuff like 2^n numbers. In a different type of proof, one can show that the sequence always has to come to a power of 2. Also a study of the longest chains of consecutive odds..whether the numbers that trigger them have any special properties or whether they are just random (and a little lucky). 
April 6th, 2017, 12:24 AM  #48 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,518 Thanks: 506 Math Focus: Yet to find out.  
April 6th, 2017, 12:43 AM  #49 
Senior Member Joined: Mar 2017 From: . Posts: 275 Thanks: 5 Math Focus: Number theory 
Joppy, we have shown that... It is impossible to have any positive integer from which the sequence would infinitely rise either due to generating an infinite consecutive odd numbers or having more odd numbers than even numbers being generated And also the sequence must always converge. It holds and the proof is rigorous. 
April 6th, 2017, 08:56 AM  #50 
Global Moderator Joined: Dec 2006 Posts: 18,593 Thanks: 1491  

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