Wikipedia:Reference desk/Archives/Mathematics/2012 January 3

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

January 3

Dowsing

James Randi did an investigation into the dowsing phenomenon:

• Water dowsing trials: 50
• probability of success in each: 0.1 (10%)
• total successes: 11 (22%)
• here is a calculator

Then Randi Suggests this can entirely be explained (away) by chance arguing there is no phenomenon. This doesn't seem correct but Wikipedia:Synth says: Do not combine material from multiple sources to reach or imply a conclusion not explicitly stated by any of the sources. I wonder if that means we cant say it is wrong? 11 in stead of 5 seems obvious enough? Just removing the source doesn't seem a good idea in this case.

• See: talk:Dowsing#foo

84.106.26.81 (talk) 05:53, 3 January 2012 (UTC)

I just did an investigation into making coin flips turn out heads by eating a pepperoni before flipping:

• Coin Flip Trials: 50
• Chance of Success in each: 0.5 (50%)
• Total Successes: 39(78%)
• here is a calculator

OMG! That's more unlikely than the above!!!!! Forget the reference desk, I'm buying a bus ticket to Vegas and a giant stack of pepperoni...Phoenixia1177 (talk) 11:18, 3 January 2012 (UTC)

Rudeness aside, 50 trials done once isn't enough to prove anything, even if the odds of such a thing happening was 1 in 1,000,000, that doesn't mean this wasn't the experiment that happened to be the one. Now, if you came and said that 500 such experiments were done with the same results at different locations by different people, that would be a lot more impacting. Even if there is something to explain here, it doesn't prove that dowsing works is the right explanation; maybe Randi is secretly a believer in dowsing and decided to slightly screw up (just to send a message to all true believers), maybe a cloud elf was moving the dowsing rod (they are full of mischief), maybe etc. But, since you only have one experiment with only 50 trials, who knows? And that's supposing that there really is something requiring an explanation.

By the way, on an aside: the problem, to me, with half of this stuff is what people infer out of it. How do you know that there aren't cloud elves pushing rods as a joke and it just happens that water inspires them to push them the right way when it is present? Real science is not finding isolated garbage, its building theories and discussing mechanisms behind things; I'd be much more likely to listen to dowsers if they gave a realistic set of mechanisms behind the supposed effect, then tested for the existence of all of those mechanisms is a variety of ways (not just by checking for water...) Anyways, cheers to you and your faith that someone wants to suppress this valuable "find" that dowsing is highly unlikely to detect water barely above chance:-) Phoenixia1177 (talk) 11:31, 3 January 2012 (UTC)

The hypothesis that you are telling the truth about getting 39 heads out of 50 fair coin tosses is rejected at a significance level of p < 0.0001. Looie496 (talk) 15:48, 3 January 2012 (UTC)

Were you being humorous or serious? If the latter, the whole point is that it doesn't really matter. If I said I got 50 out of 50, it wouldn't matter. That doesn't really prove dowsing, it would prove something that might be interesting maybe occurred and might have effected things, etc on the "maybes". All this talking about probability misses the whole point, if one small experiment got it all right, I still wouldn't be impressed; so I'm certainly not impressed with the one's given. Dowsing, cloud elves, wizards playing pranks, luck, error, fraud, etc. They all support it just as well. Give me a theory of of dowsing, then test each of these mechanisms four or five ways in several highly controlled experiments with a large number of trials, then there'd be something worth discussing. Too, as for the probability thing, when playing poker, if someone gets a royal flush, I'm going to scoop up all my money from the pot and run home declaring them a cheater...Phoenixia1177 (talk) 04:39, 4 January 2012 (UTC)

By the way, I get that one can say dowsing is just the event happening itself sans explanations, but this is not what dowsers, at least that I've read about, seem to be saying. But, even you were saying that, this one test wouldn't demonstrate to me, it could still be error, it could still be fraud, it could still be a bunch of other stuff. So, even if you were to insist that we were just looking into the existence of a phenomena, it would still require more than a small number of trials from one test; which brings me back to wondering why we are even discussing the probability of this one test's results since it doesn't really establish anything. If you disagree, would you take medicine that had only been tested in one trial unless it was a last ditch effort? I wouldn't. Phoenixia1177 (talk) 11:29, 4 January 2012 (UTC)

The trial produced 15 out of 111 which is just random. On could have chosen the 4 out of 61 in the remaining part and said one should avoid digging in the places where the dowsers indicated! Dmcq (talk) 12:13, 3 January 2012 (UTC)

If there were k=11 hits out of n=50 trials, what is the probability P of making a hit? The answer is that we cannot tell for sure, but P can be estimated by the mean value μ = (k+1)/(n+2) = 0.230769 give or take the standard deviation σ = √(μ(1−μ)/(n+3)) = 0.0578734. (See this.) The claimed value 0.1 is (μ−0.1)/σ = 2.25957 standard deviations below the mean value. So the hypothesis P=0.1 can not definitely be rejected. The binomial calculator is not relevant. Bo Jacoby (talk) 12:28, 3 January 2012 (UTC).

And, based on Dmcq's comment, this is after cherry-picking. I won't be in too much of a hurry to get a dowsing rod, then. Does anyone have a link to the report of the original trial? -- The Anome (talk) 12:41, 3 January 2012 (UTC)

Model Theory and Category Theory

Hi, random musing led me to this question, apologize if it is fairly obvious or fairly nonsense. Anyways: is there some logic (1st order, 2nd order, etc.) so that for every category C there is a structure S so that the category of models of S has C as a full subcategory? More generally, are there any results looking at the extent to which categories can be represented/related to the classes of models for structures (not necc. 1st order)? Thank you for any help:-) Phoenixia1177 (talk) 10:41, 3 January 2012 (UTC)

Let's see if I understand your question. You're asking if does there exist a logic such that for every category C there is there sentences in that logic where the categry C embedds into the category of models (probably mod isomorphism) of those sentences where arrows are "elementary" embeddings (just thinking first order). Let's stay first order (this problem will exist anywhere where the models are sets). I can see a problem with a category class big and every object pointing to that object. That will imply that the object pointed to (something like a monster model) is class-big. If your category is locally small, then I'd say this must be true in first order model theory; just with models of the empty set even. Those are the extremes though. Wgunther (talk) 15:30, 3 January 2012 (UTC)

Thank you for your response:-) I don't understand the problem about a proper class with every object having an arrow pointing some given object. SETS is a proper class and there is an arrow between any two objects; the same goes for several other categories. The best possible objection I can come up with is that you could control the semigroup structure of endomorphism's too broadly and end up getting situations not possible in a model. For example, take a category and theory that works, for the catergory:delete one endomorphism, not id, for each object, then select a maximal subsemigroup and use that as the collection of endomorphisms; I would think that you could use the properties of the working pair to get an argument showing that no theory could exist for the new category unless it had such and such relations to the original, then show that there could be no such set of sentences. Or some such. Please excuse the bad writing skills, I'm at work. Phoenixia1177 (talk) 04:57, 4 January 2012 (UTC)

Differentials and uncertainties

Hello. I am trying to calculate the uncertainty in my k (Hooke's constant) for the formula k=F/x (Hooke's law, scalar form), from the values and uncertainties in the F and x. I'm using a differential approximation for the uncertainty and the quotient rule. My reasoning is: $k={\frac {F}{x}}\implies dk={\frac {x\cdot dF-F\cdot dx}{dx^{2}}}$ I have that dF=±0.02 N, dx=±0.1 cm, F=0.50 N, x=37.0 cm, however I'm getting an unacceptably huge uncertainty dk (=±69 N/cm!). Am I doing something wrong? Thanks. 24.92.85.35 (talk) 19:38, 3 January 2012 (UTC)

The denominator is x², not dx². Bo Jacoby (talk) 19:41, 3 January 2012 (UTC).

Ah, of course! I was just thinking the differentials didn't match up, thank you! 24.92.85.35 (talk) 20:29, 3 January 2012 (UTC)

You are welcome. By the way, you are probably interested in the relative uncertainty dk/k rather than the absolute uncertainty dk. When k = F/x then log(k) = log(F)−log(x) ; d(log(k)) = d(log(F))−d(log(x)) ; dk/k = dF/F−dx/x = ±0.02/0.50±0.1/37 = ±0.04±0.0027 = ±√(0.04²+0.0027²) = ±0.0401 ≈ ±0.04. So k=1.35·e^±0.04N/m. Bo Jacoby (talk) 04:52, 4 January 2012 (UTC).