Wikipedia:Reference desk/Archives/Mathematics/2015 March 28

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March 28

Chances of hitting triple 20 on a dartboard

Assume that your throws on a dartboard follow a circular Gaussian with standard deviation σ, centered on the middle of the triple 20 region. What are your chances of hitting triple 20? Has anybody made a graph or table of this? --98.232.12.250 (talk) 09:12, 28 March 2015 (UTC)

If the region is a disc of radius r, and the x and y coordinates are independently normal with sd $\sigma$, then the probability is $1-\exp \left({\frac {-r^{2}}{2\sigma ^{2}}}\right)$. -- Meni Rosenfeld (talk) 18:02, 28 March 2015 (UTC)

The triple 20 region isn't a disk. It's a sector of a circular ring. – b_jonas 18:34, 28 March 2015 (UTC)

With the assumptions listed by Meni Rosenfeld above, the distance $r$ from the center follows the Rayleigh distribution with parameter $\sigma$, while the angle is uniformly distributed. Hence the chance for hitting a triple-20 is

${\frac {1}{20}}\left[\exp \left({\frac {-r_{\text{in}}^{2}}{2\sigma ^{2}}}\right)-\exp \left({\frac {-r_{\text{out}}^{2}}{2\sigma ^{2}}}\right)\right]$

where $r_{\text{in}}$ and $r_{\text{out}}$ are the inner and outer radii of the ring respectively. Abecedare (talk) 20:56, 28 March 2015 (UTC)

This would have been true if the dart was aimed at the center of the board. I assumed this to be the case, since I assumed the target region is in the center; however, now that I see what region we are really talking about, the OP's specification "centered on the middle of the triple 20 region" is crucial, and complicates things. It should be easy to find numerically, given the radii. -- Meni Rosenfeld (talk) 23:12, 28 March 2015 (UTC)

One would still need to define "the middle of the triple 20 region" precisely. Would this be the region's centroid, or a point midway between the inner and outer arcs? —Quondum 23:20, 28 March 2015 (UTC)

(edit conflict)True. I had missed the "centered on the middle of the triple 20 region" in the question. Its trivial to write the integral expression for the solution, but I too doubt that it will result in a closed form expression, or at least one that is easily interpretable. Numerical computation would be the way to go.

Interesting side question: what is the optimal point within the triple 20 region to aim at? Center of mass? At mean radii? My guess is neither. Abecedare (talk) 23:27, 28 March 2015 (UTC)

I really should do something about my nerd sniping problem.

Anyway, I've found online that the relevant radii are $(9.525,10.478)\ \mathrm {cm}$. Assuming $\sigma =1\mathrm {cm}$, the optimal target radius is 9.97541 cm, somewhere between the centroid radius of 9.96796 and the mid radius of 10.0015 cm (but closer to the centroid). The chance is then 32.3883%.

If we assume for simplicity we're aiming at the centroid, and varying $\sigma$, a graph of the hitting probability is at http://i.imgur.com/3bUIAwf.png. For large $\sigma$, this is ${\frac {2.99439}{2\pi \sigma ^{2}}}.$

Mathematica code:

f[a_, \[Sigma]_] := NIntegrate[ r Exp[(-(r Cos[\[Theta]] - a)^2 - (r Sin[\[Theta]])^2)/( 2 \[Sigma]^2)]/(2 \[Pi] \[Sigma]^2), {r, 9525/1000, 10478/1000}, {\[Theta], -\[Pi]/20, \[Pi]/20}]

-- Meni Rosenfeld (talk) 13:13, 29 March 2015 (UTC)

"The chance is then 32.4988%". Personally that represents my chance of hitting the board.  — An optimist on the run! (logged on as Pek the Penguin) 12:11, 30 March 2015 (UTC)

Why $\tan \pi /20$? Shouldn't the limits on angle just be $\pm \pi /20$? --Tardis (talk) 03:37, 1 April 2015 (UTC)

You're absolutely right, that's a relic of an earlier calculation that attempted to define the region using the ratio of y and x. (Part of the reason I post code is so that I can get called on errors like this).

Thankfully, this doesn't change the results by more than ~1%. I'll try to post corrected results later. -- Meni Rosenfeld (talk) 09:14, 1 April 2015 (UTC)

Nullity

In the transreal numbers, can anyone explain the kind of number nullity is?? Any examples of mathematical equations that equal nullity?? Georgia guy (talk) 23:39, 28 March 2015 (UTC)

A nonsense number? :-Þ Double sharp (talk) 03:39, 29 March 2015 (UTC)

Seriously, he claims that 0 ÷ 0 is nullity, which means that 0 = 0 × nullity. But in fact 0 = 0x is true for all real numbers x, which is precisely why this expression is called "indeterminate", so his definition kinda misses the point. It also breaks most of the identities, forcing you to add silly exceptions like "x/x = 1, except 0/0 = nullity". Anderson explains in a paper that the problem with NaN is that NaN ≠ NaN, which shouldn't occur for a number: hence his "nullity" idea. Of course, he doesn't seem to notice that NaN is by definition not a number, so why should you expect concepts like comparison and arithmetic to work on it when they only work on numbers? Apart from that, his nullity behaves exactly the same way as NaN, which ought to be a big hint about its nature.

According to his definition, 0 ÷ 0 = 0⁰ = nullity. Both would rather better be left as indeterminate forms (except in the instances when 0⁰ = 1 is a convenient definition). (And also, the choice of the word "nullity" is a bad idea: it already has several meanings in actual mathematics!) Double sharp (talk) 03:47, 29 March 2015 (UTC)

P.S. Oh, I can't believe I've forgotten to add this: adding nullity also breaks the field structure of the real numbers. That is not good. (And yet he implicitly uses properties stemming from this field structure in his proof that 0/0 = 0⁰ = nullity. Go figure.) Double sharp (talk) 03:58, 29 March 2015 (UTC)

I've being playing with an arithmetic system based on something Sławomir mused about on this talk page not long ago. There is a kind of logic in it. In particular, one can extend the real numbers consistently to include ∞ and nullity (Φ). Unlike Anderson's idea, +∞ = −∞, but otherwise it essentially the same. The attractive bit is that there is a homomorphism from an easily understood ring-like object to this extension of the reals, and it seems to generalize to any field. I have not considered expressions such as 0⁰ yet. And as to "silly exceptions", disallowing x ÷ 0 may be regarded as such; with this extension, one is replacing exceptions with operations that do not universally solve to other operations (e.g. one gains that 1 ÷ 0 is defined at the cost of the reciprocal and multiplicative inverse of a number are occasionally not the same thing, but only when the latter does not exist anyway). —Quondum 04:36, 29 March 2015 (UTC)

I see that this is already covered by Wheel theory. —Quondum 04:53, 29 March 2015 (UTC)

0⁰ = 1 isn't just a "convenient definition". It's an essential truth, if we're talking about the natural number 0 in the exponent, rather than the real number. Natural exponentiation is defined as repeated multiplication. $x^{0}$ is an empty product, equal to the multiplicative identity. It doesn't matter if the nonexistent multiplicands are 0 or anything else, because there are none of them. -- Meni Rosenfeld (talk) 12:00, 29 March 2015 (UTC)

You can go further. xⁿ would be defined naturally in terms of repeated operations for all x of a wheel and all integers n. A continuous version of exponentiation would presumably still suffer problems though. —Quondum 15:43, 29 March 2015 (UTC)

A good rule of thumb is this: when someone you've never heard of, with no clear academic history in a particular discipline, starts to assert that he (and it is almost invariably he) has solved some outstanding problem in this discipline, or (as is the case here) has "solved" something that appears to be a "problem" in this discipline to the lay man, it is almost always the case that you are dealing with a crank.

If, in addition, he starts claiming (as in this case) that he has been able to solve outstanding problems in a whole range of only peripherally-related fields ("He has also claimed that it can help solve such problems as quantum gravity, the mind-body connection, consciousness and free will.") you are dealing with an egomaniacal crank who has progressed to full-blown nutter status.

Finally, although I haven't wasted much time reading his system, it strikes me as immediately problematic. He defines only one kind of infinity. This suggests to me an immediate problem, as he is unable to answer questions about the cardinality of the power set of a set whose cardinality is his "infinity". I imagine a more detailed inspection will reveal other problems, as some earlier answers already suggest. RomanSpa (talk) 10:12, 29 March 2015 (UTC)

Just to emphasize - Anderson, who has coined the term "nullity" in this context, is well-known as a crank. Don't expect anyone to be able to explain his ramblings. However, there is real mathematics dealing with concepts similar to those he has presented, see the other replies here for pointers. -- Meni Rosenfeld (talk) 12:00, 29 March 2015 (UTC)

And then you've got Inter-universal Teichmüller theory which not only breaks the rule of thumb, it probably breaks the thumbs...Naraht (talk) 18:06, 30 March 2015 (UTC)

Mochizuki isn't "someone [mathematicians have] never heard of, with no clear academic history". -- BenRG (talk) 20:58, 30 March 2015 (UTC)

True, which makes things even more frustrating.Naraht (talk) 20:05, 31 March 2015 (UTC)