Wikipedia:Reference desk/Archives/Mathematics/2012 October 6

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October 6

Root problem

Can someone explain for me how to do problem 3? I have the answer key but unable to understand what is it saying. I don't get how (p^2 - 1)/4 is the product of the roots. And the part where it say "which checks vs. the coefficient of the x term" --> What is this even mean? 65.128.190.136 (talk) 03:05, 6 October 2012 (UTC)

They are talking about simplifying this expression:

x2 - px + (p2-1)/4

into these roots:

[x + (p+1)/2] [x + (p-1)/2]  (See correction below.)

The first step is to resolve:

(p2-1)/4

which can be rewritten as:

p2/4-1/4

into it's roots:

[(p/2)+(1/2)] [(p/2)-(1/2)]

or:

[(p+1)/2] [(p-1)/2]

You might recognize this as a difference of squares problem, where (a+b)(a-b) = a² - b². StuRat (talk) 03:35, 6 October 2012 (UTC)

Why when I multiplied this out. I got everything the same except x^2 + px instead of - px. How did they get -px? Can you show me? Thanks!65.128.190.136 (talk) 03:57, 6 October 2012 (UTC)

Oops, I had the signs wrong on the roots. Here's the corrected version:

[x - (p+1)/2] [x - (p-1)/2]

StuRat (talk) 04:11, 6 October 2012 (UTC)

Okie! I think everything is clear! Thanks!65.128.190.136 (talk) 04:20, 6 October 2012 (UTC)

Your quite welcome. I'll mark this Q resolved. StuRat (talk) 04:33, 6 October 2012 (UTC)

Resolved

Relations: any usual way to denote/describe the following relation?

1. Let R be a relation. Is there a better way, to denote (mathematically by R), and/or describe (verbally by R), the following relations?

• { (a,(b,c)) | ((a,b),c)∈R }
• { ((a,c),(b,d)) | ((a,b),(c,d))∈R }

.

2. If R and S are relations, then the relation:

• {(a,c) | there exists b, such that (a,b) ∈ R and (b,c) ∈ S}

is usually-denoted as S∘R, and is usually-described (verbally) as the composition of S and R.

How about the following relation?

• { ((a,c),b) | (a,b)∈R and (b,c)∈S }

Is there a better way to denote it (mathematically by R,S), and/or describe it (verbally by R,S)?

84.229.139.122 (talk) 18:01, 6 October 2012 (UTC)

.

No. Looie496 (talk) 23:44, 6 October 2012 (UTC)

Not sure what would be considered "better", but writing the relation as a function seems a bit clearer to me. if S is f(x) and R is g(x) then S∘R is f(g(x)). Ssscienccce (talk) 00:16, 9 October 2012 (UTC)

I'm not asking about the relation denoted by S∘R. Please review the exact relation I'm talking about. I've just emphasized some words by bold letters, in order to make this clearer. 84.229.139.122 (talk) 07:18, 9 October 2012 (UTC)

Statistically, is this a valid argument?

A common argument for the existence of life on other worlds in our universe is that there are just so many planets out there that there that it would extraordinarily unlikely for life to not evolve on at least some of them. Basically, this argument is making the claim that it is very unlikely that the probability of life emerging on a planet, p, is very low. But is this reasoning valid? Is it really possible to justify the claim that it's more unlikely that p < 10^-20 than p > 10^-20? 74.15.136.9 (talk) 20:42, 6 October 2012 (UTC)

No, that's a weak argument, especially when accompanied by "it evolved here, so it can't be that unlikely". The problem is that only on planets with intelligent life can they know that life has evolved there, so, the observed rate on your own planet will always be 100%, no matter how low the overall rate is. Now, if we had evidence that life had evolved independently several times on Earth, that would be different. Life does seem to have evolved quite early on Earth, and that would support the idea that it's not all that rare. However, again, had it evolved a billion years later, there would be nobody around to observe it yet, so our observation is again biased. StuRat (talk) 21:00, 6 October 2012 (UTC)

Also note that, according to the many worlds hypothesis, there may be an infinite number of planets. If so, then, no matter how rare the existence of life is, there must not only be more than one, but an infinite number of worlds with life. This doesn't mean that any are accessible from Earth, however. StuRat (talk) 21:10, 6 October 2012 (UTC)

The OP specified that (s)he was only talking about our universe. Rojomoke (talk) 22:28, 6 October 2012 (UTC)

Well, that depends on your definition of "universe". I'm using def 1 under Wikt:universe, and you are using def 2. Note that the etymology supports def 1. StuRat (talk) 22:39, 6 October 2012 (UTC)

The universe (in the ordinary sense of the word) may well be infinite anyway, so I don't see why you need "many worlds". --Trovatore (talk) 00:34, 13 October 2012 (UTC)

I'm not following. The amount of mass in our Big Bang must be finite, I believe, or the universe would have quickly collapsed back into a black hole. So, for the universe to be infinite, other than the trivial case of infinite nothingness, there must be an infinite number of Big Bangs (or colliding branes, or what have you). Isn't this, by definition, a multi-verse ? StuRat (talk) 03:58, 14 October 2012 (UTC)

• This issue is often discussed in terms of the Drake equation, which helps to make the numerics more specific. I think you might find our article on it interesting. Looie496 (talk) 23:42, 6 October 2012 (UTC)

• The really unlikely scenario is that life evolved on three planets, or ten, or some other small number. Most probably, either life is unique to Earth, or it has evolved independently billions of times in the Universe. 86.160.216.227 (talk) 00:29, 7 October 2012 (UTC)

On a related point Nick Lane, in his bookPower, Sex, Suicide: Mitochondria and the Meaning of Life argues that evolution of multicellular lifeforms is exceedingly unlikely (this is not the question of any life evolving, though), should this be of interest. The point of life evolving being either very likely or very unlikely is a good one, and supports the point that the originally questioned reasoning has no merit. — Quondum 12:53, 7 October 2012 (UTC)

Let p = the probability of life forming on a given planet, and let the planets be independent as to whether or not they support life. (Or if you don't like that independence assumption for two planets in the same star system, let p be the probability that a star system supports life, with each star system independent.) N is the number of planets. Before observing any planets (even Earth), the number of planets with life is binomially distributed with mean pN. If we hypothesize that p is much smaller than 1/N, then the probability of there being even one planet with life is very small, and hence not consistent with the observation that at least one does in fact exist. So a maximum likelihood argument for getting an estimate of p, conditional on knowing #planets with life >0, will not choose an estimate for p that is very much below 1/N. So let's make the conservative assumption that p = 1/N rather than p>1/N. Now, what is the probability that with N-1 planets that have not been monitored for life, none of them support life? The probability that any one of those planets does not support life is 1-p = 1-(1/N). The probability that all N-1 of them do not support life is [1-(1/N)]^(N-1). We could set N = 10^11 (a conservative estimate of the number of stars in our galaxy and assuming an average of one planet per star -- use your own estimate for the number of planets if you prefer), but my calculator can't deal with that. But if I assume one billion planets, N=10^9, I get a probability of .3679 that no other planets have life. I think if you let N go up from there by a couple orders of magnitude you'd get a much lower probability. Duoduoduo (talk) 14:00, 7 October 2012 (UTC)

The observation that we exist that you want to condition on has to be taken into account in the correct way. This is a non-trivial issue, see here. See also the sleeping Beauty problem. Count Iblis (talk) 15:44, 7 October 2012 (UTC)

The way I'd look at it is to get intelligent life we have to get a suitable planet, then life has to arise, then complex multicellular life, then intelligence. If we are the only intelligent life in the universe then the ratios of the probabilities of each of these happing per unit time by maximum likelihood is I think proportional to about 1/(time it took to happen). They seem to be in the proportion about 8 billion to perhaps a fraction of a billion to 2 billion to 2 billion. It seems that for a suitable planet lie will arise pretty quickly. The interesting thing is that the other factors are of about the same order of magnitude and the least likely to happen per unit time is a suitable planet for life. If there was one stage which was very difficult I would expect it to dominate the others so for instance if intelligence was the hard thing we'd have had multicellular life in less than a billion years and have spent the next 4 billion waiting for intelligent life. This argument does unfortunately give the chancs of there being life elsewhere, only ratios of probabilities, but having no relatively small probability means to me that there i no particular reason forus to be the only intelligent life - it is quite possible there are others just they are in other galaxies and if we wait around a few more billion very possibly we'll see some in our own galaxy - and there is very likely to be life already in this galaxy. Dmcq (talk) 00:30, 8 October 2012 (UTC)

I think this thread is diverging from the original question, which seems to be asking whether it is valid to argue that life has at least some significant probability as with the Drake equation within the field of statistics. And I think that the correct answer is that it has no validity as a statistical argument; it is merely a plausibility argument given a number of (statistically) untested conjectures. For example, if you start with the given that we have one life-supporting planet, it would be false to conclude on statistical grounds that the probability of this arising is non-vanishing. Thus, simply put, the argument is invalid from a statistical perspective. Additionally, what observational evidence we do have tends to support (also using the plausibility approach) the idea that treating the one known instance of a life-bearing planet as a representative sample leads to a wildly high estimate of the probability of life arising. — Quondum 04:44, 8 October 2012 (UTC)

All statistical results are based on Bayesian priors, it is probability theory that doesn't need assumptions as it is the mathematical idealization. I think there is quite enough evidence to derive reasonable statistics about life with plausible assumptions, as I pointed out above the ratios of the various timescales indicates there is no particular sticking point. The major problem is the number of planets that are suitable for life in a galaxy like ours and how often life is totally wiped out by something like a huge meteor or local supernova. I don't think we can say anything much about what has happened in other galaxies though I guess in the future we could have a galactic scale SETI program. Dmcq (talk) 00:21, 9 October 2012 (UTC)