[seqfan] Re: Probability of identical sequences

[David Seal]

> If we have two sequences, A and B, with different definitions. However,
> when we calculate k terms for each sequence, all of these terms are
> identical. If we can’t prove that definition A equals to definition B,
> what’s the probability that A and B are identical?

It's not defined - you need to specify the distribution of the sequence definitions for the probability to exist.

An analogy: what is the probability that two numbers in the range 0-5 are the same? The answer is that it's undefined, because the question doesn't specify the distribution of the numbers. One could specify it in the question, and then the answer would be defined - two possible forms of that question are:

* "What is the probability that two numbers in the range 0-5 generated by throwing a fair die and subtracting 1 from the result are the same?" The answer to that question is (1/6)^2 + (1/6)^2 + (1/6)^2 + (1/6)^2 + (1/6)^2 + (1/6)^2 = 0.16666...

* "What is the probability that two numbers in the range 0-5 generated by tossing five fair coins and counting the number of heads are the same?" The answer to that question is (1/32)^2 + (5/32)^2 + (10/32)^2 + (10/32)^2 + (5/32)^2 + (1/32)^2 = 0.24609375.

Clearly with the answer depending on which of those two distributions of the numbers (or many other possibilities) is being asked about, it's going to be undefined if the question doesn't specify the distribution. And unfortunately, assuming one wants the distribution to have the property that for any reasonable outcome, it has a nonzero chance of generating that outcome, it's a lot harder to specify a well-defined distribution of sequence definitions than a well-defined distribution of numbers in the range 0-5.

David


> On 13 January 2020 at 19:06 Ali Sada via SeqFan <seqfan at list.seqfan.eu> wrote:
> 
> 
> Hi Everyone, 
> 
> If we have two sequences, A and B, with different definitions. However, when we calculate k terms for each sequence, all of these terms are identical. If we can’t prove that definition A equals to definition B, what’s the probability that A and B are identical? Is it zero, since we are dealing with infinite terms? Or is it a function of k?  
> 
> This question came to me when I was trying this algorithm: Exchange n and 2n. Each term gets changed only once. 
> a(1)=2 and a(2)=1.
> a(3)=6 and a(6)=3
> etc.
> The sequence I got was 
> 2, 1, 6, 8, 10, 3, 14, 4, 18, 5, 22, 24, 26, 7, 30, 32, 34, 9, 38, 40, 42, 11, 46, 12, 50, 13
> 
> This sequence seems identical to A073675 (Rearrangement of natural numbers such that a(n) is the smallest proper divisor of n not included earlier but if no such divisor exists then a(n) is the smallest proper multiple of n not included earlier, subject always to the condition that a(n) is not equal to n.)
> 
> This example might not reflect my question above exactly. I am just trying to show why I asked the question.
> 
> Best,
> 
> Ali
> 
> 
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/