Wikipedia:Reference desk/Archives/Mathematics/2007 August 28

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August 28

numbers odds/fund raising raffle

There is a large fund raising raffle in my area that, for a price of a ticket, you get 2 (two) sets of 4 numbers as your numbers for the raffle.For instance, 3520 and 9940, and at the raffle site, the organizers use bingo drums to randomly pull out 4 drawn numbers for that particular prize. Your two sets of numbers are yours for the entire event. Every 5 minutes, they spin the bingo drum, and yell out 4 new numbers. I may be wrong, but I can recall only a few winners that were present at the event the last 4 years. What do you suppose the odds are winning in a format such as this? They use #'s 0 thru 9 and throw the pulled ball back in the drum for each spin so in reality you could also have four of a kind pulled. Great fund raiser....beer & food supplied but the ODDS!!!! Bigbill. —Preceding unsigned comment added by Bigbill1130 (talk • contribs) 01:37, August 28, 2007 (UTC)

If they have all numbers from 0000 to 9999 in the bingo drums, that would give each number a 1/10000 chance, or approximately a 1/5000 chance for two numbers, per draw. If there are 10 draws then that would be approximately a 1/500 chance of winning. If, however, they only put balls in with the numbers which have been sold, then the odds are better, depending on how many were sold. StuRat 02:36, 28 August 2007 (UTC)

I understand the question to mean that the bingo drum contains a number of balls (or whatever) with single digits on them. A selection of 4 random balls constitutes a single 4-digit number. That would make the odds somewhat different. -- JackofOz 02:42, 28 August 2007 (UTC)

In that case, the odds are exactly the same as in my first example, where there are balls numbered 0000 to 9999 (if using one bingo drum, this assumes they return the balls after each digit is drawn; failure to do so would mean only 4 digit numbers with all unique digits would be possible). StuRat 11:29, 28 August 2007 (UTC)

The tickets that you purchase have 2 sets of numbers( four digits each) just randomly printed on them.I did ask one of the volunteers concerning how many tickets were sold and how many tickets were printed for the event. Zillions & Zillions was the answer and he walked away! —Preceding unsigned comment added by Bigbill1130 (talk • contribs) 11:42, 28 August 2007

The poster states that the numbers are 0-9, replaced after each draw, so the case is as your first analysis. Just a note for those not so familiar with probability: StuRat notes that if you have a 1/5000 chance to win, and you draw 10 times, your total chance to win is about 1/500. This is true; however, your chance is not exactly 1/500 (it's right around 1/500.45). Similarly, if you flip a coin (chance to get heads: 1/2) and you flip it twice, the chance of getting heads at least once is not 1, since you could get tails twice. The formula is $1-(1-{\frac {1}{5000}})^{10}$, which is close to 1/500 but not exactly equal. (I couldn't find an article addressing this, but we must have one...) Tesseran 20:03, 28 August 2007 (UTC)

That would be why I said "approximately". I gauged the poster's apparent skill level and decided he wouldn't appreciate doing the extra math for the difference between the 1/500 estimate and 1/500.45 reality. But, I suppose some other readers might want the exact answer. StuRat 01:53, 30 August 2007 (UTC)

It would take 4 hours and 15 minutes (51 draws) to reach the level of 1% chance of winning a prize.  --Lambiam 19:52, 28 August 2007 (UTC)

More Confusion About Infinity

In reading responses to my question about Infinity (above), I thought of another question / puzzle / dilemma that I don't quite understand. If someone can break this down into simplistic terms, so that I can understand it, that would be great. Thanks. Let's say that (with a pencil) I draw a line segment AB and it is, say, exactly one inch in length. The graphic illustration would be a point/dot "A" that is connected by a straight line (segment) to a point/dot "B" ... that line segment would measure exactly 1 inch. Just as as illustration, let's just say that the "A" represents the number 7 on a number line or the 7-inch tick on a ruler ... and that the "B" represents the number 8 on a number line or the 8-inch tick on a ruler. Now, between the number 7 and the number 8, there are an infinite amount of other numbers. If we kept cutting the line segment in halves, we would find the tick mark on a ruler for 7.5, then 7.25, then 7.125, then 7.0625, etc. etc. etc. This process would never end, as there are an infinite amount of numbers between 7.000 and 8.000. We can keep halving forever. How is it, then, that when I physically draw that line segment AB, there actually is a finite amount of points / dots in between the A and the B? Stated in other words ... how can an infinite quantity of numbers be accurately represented in a physical drawing/ illustration with a finite amount of points/ dots? I hope my question is making sense. That is, realistically, if I wanted to draw a pencil-point "dot" for every single value between 7.00 and 8.00, I would be drawing forever and ever and ever -- I would never finish. Because there are an infinite amount of numbers (and, thus, an infinite amount of pencil-point dots to be drawn) to accurately represent the "distance" (all the tiny dots) from the "A" point to the "B" point. How, then, is it possible to draw a line (that is, a connection of dot after dot after dot) that is exactly one inch in length? On some level, shouldn't that be an impossible (i.e., infinite, unending) task? Perhaps stated differently: If I wanted to accurately draw a line segment of exactly one inch on a ruler that goes from the 7-inch tick mark to the 8-inch tick mark ... I would draw the dot for 7 (that's "A") ... then I would draw the dot for 8 (that's "B") ... then I would draw the dot that represents the 7.5 ... then the dot that represents 7.25 ... then the dot that represents 7.125 ... then the dot that represents 7.0625 ... and so on. I could never possibly finish this process of drawing all these dots. How, then, do I -- in fact -- actually come to be able to "draw" this one-inch succession of infinite dots (without actually drawing an infinite amount of dots)? Aaaaaaaaaaaaarrrrrrrrrrrgggggggggghhhhhhhhhhhhhhhh! (Joseph A. Spadaro 06:29, 28 August 2007 (UTC))

The simple answer is that the number line is an abstraction, and has certain properties (such as infinite divisibility) that are not shared by any physical line. So a physcial line, no matter how finely it is drawn, can only be an approximation to the "true" number line. You might be interested in reading about Zeno's paradoxes, and some of the proposed "solutions" to them. If you wonder about philosphical questions such as "if an abstract concept is supposed to have properties that no physical object can have, how can we be sure that our reasoning about the abstract concept is correct ?" then you might find our article on idealism interesting. Gandalf61 08:16, 28 August 2007 (UTC)

The first thing I thought of when I read that post were Zeno's paradoxes. Joseph A. Spadaro's paradox is like those of Zeno. A.Z. 05:23, 3 September 2007 (UTC)

Instead of saying that a physical line only approximates the number line, you can say that the number line only approximates a physical line. In calculus one normally thinks of discrete sums as successive approximations to the integral, but in the sciences one often uses the integral as an approximation to a truly discrete sum. -- BenRG 21:23, 28 August 2007 (UTC)

The more simple answer is that the pencil dot has a finite diameter. So when making a dot you mark some interval rather than a number or point. If a dot is 1/100 inch thick, it's enough to mark a hundred dots on a line to make a continuous, 1 inch long line representing [7, 8] interval. However, it is not the same as a hundred points, because an (ideal) point has no width—so there is infinitely many points between any two points on a line, while there's only finite number of distinguishable dots you can make on a 1-inch line. --CiaPan 09:11, 28 August 2007 (UTC)

Agreed, mathematical points have zero height, width, or length, while the real-world dots that make up a real-world line (say from a dot-matrix printer), do have finite dimensions. A pencil-drawn line, if viewed on a microscopic level, would consist of lumps of graphite with finite dimensions. Even at the smallest possible level, they would consist of carbon atoms, which have an extremely small, but still finite, diameter. StuRat 11:24, 28 August 2007 (UTC)

And think about this: lines have no thickness, so you have never even seen a line in your whole life! –King Bee ^{(τ • γ)} 11:02, 28 August 2007 (UTC)

Another view. Suppose the tip of the pen is a single point. Since the pen is moving, every new instant of time corresponds to a new pen position on the paper. Then one could say that the infinite number of points drawn on your paper comes from the infinite number of instants in the time interval used. One might also want to note that just as sweeping a point generates a line, sweeping a line generates a surface (an infinite number of lines), sweeping a surface generates a volume etc. —Bromskloss 11:52, 28 August 2007 (UTC)

One possible explanation (though perhaps a science desk answer) would be that as you draw with the pencil from A to B you are rubbing of graphite 'atoms' which have finite size - therefor the max number of 'dots' is equal to the number of 'atoms' you have rubbed off the tip of your pencil. Is that a good answer or have I misunderstod (perhaps you don't accept the "atomic principle" if so you are back to square one - see below)?87.102.90.8 14:07, 28 August 2007 (UTC)

If you think matter is continuosly divisible down to zero size then in fact you will have drawn infinite dots - a continuous line even at the microscopic level..87.102.90.8 14:07, 28 August 2007 (UTC)

Fine, but it isn't. See planck length. —Preceding unsigned comment added by 129.78.64.102 (talk) 03:21, August 29, 2007 (UTC)

Point - if you think an article/rule/measurement that is based within the belief/construct that things are quantised proves that the construct is true then you are wrong! think about it...87.102.18.14 11:19, 29 August 2007 (UTC)

...which is where we came in. It is simply not possible to create an exact physical model of mathematical abstractions such as a point, a line, a circle etc. I've never fully understood why anyone is surprised by this. Heck, we can't even find physical models of sufficiently large integers, due to the finite size of the observable universe. There may be some interesting philosophical questions around how we can make accurate statements about the properties of such abstract "ideals" that have no physical counterparts or why they appear to help us draw useful practical conclusions about the "real world" - but most mathematicians and scientists don't spend much time worrying about this. Gandalf61 09:25, 29 August 2007 (UTC)

Can someone help me understand this integral

$\int {\cos {(\ln {(x)})}dx}$

I know that one way to do it is integration by parts, which I did and I got:

${\frac {x\cos {(\ln {(x)})}+x\sin {(\ln {(x)})}}{2}}+C$

But then when I did it another way, by multiplying the inside by ${\frac {x}{x}}$ and then using a u sub (u = ln x , du = 1/x dx) and I got:

${\frac {e^{x}\cos {(x)}+e^{x}\sin {(x)}}{2}}+C$

WRYYYYYYY?

--Funnyguy555 14:24, 28 August 2007 (UTC)

So... after the u-sub you end up with $\int {x\cos {(u)}du}$

Which would be difficult to integrate, as you've got two different variables. But if you "cheated" and did e^u = x, you get

$\int {e^{u}\cos {(u)}du}$

Which integrates to ${\frac {e^{u}\cos {(u)}+e^{u}\sin {(u)}}{2}}+C$ (note the u's)

Replacing the u's with ln x's, you get ${\frac {x\cos {(\ln {(x)})}+x\sin {(\ln {(x)})}}{2}}+C$

You just forgot to substitute back after integrating. Gscshoyru 14:34, 28 August 2007 (UTC)

This isn't cheating: if f(x)=x^2, then you can compute $\int f\,dx$ even though it has two variables—precisely because one variable depends on the other. (This is more usually written $\int f(x)\,dx$, but that's just notation.) Note that if x and y are independent variables, then the integral $\int {x\cos {(y)}dy}$ is just $x\sin(y)$, since we may treat x as a constant. This issue—keeping track of which variables are dependent on each other—is at the heart of u-substitution (also known as the chain rule), and yet it was completely glossed over in all my calculus classes. Tesseran 20:33, 28 August 2007 (UTC)

I know it isn't really cheating -- that's why I put it in quotes. It just was never taught that way in calc and when I worked out that you could do that, it felt like cheating. It made a very hard problem much easier. From then on I always sorta thought of of it as cheating. I just think for most people its a non-intuitive thing to do. Gscshoyru 22:44, 28 August 2007 (UTC)

Surface area of a drawn circle

If I draw a circle of radius r, the area of the circle is Pi r^2

However if I'm on earth and if I have to take the curvature of earth into account. Given that the radius of earth is R=6,372,797 metres. If I draw a circle on flat ground with a radius r, what is the (surface) area of the circle? For small radius of 1 metre what is the difference in surface area between Pi r^2 and the actual area? 202.168.50.40 23:59, 28 August 2007 (UTC)

See Spherical cap. When you say "radius r", that's ambiguous. Do you mean the actual radius of the circle, as measured from its center beneath the surface of the Earth, or the straight-line distance from its "center" on the Earth's surface, or the length of the great circle arc that connects the surface "center" to a point on the circle (what you would actually measure with a flexible tape measure)?

If you mean the actual radius, the surface area is $2\pi R\left(R-{\sqrt {R^{2}-r^{2}}}\right)$. If r is small compared to R, that's approximately $2\pi r^{2}\left(1+{\frac {r^{2}}{8R^{2}}}\right)$, so the difference is ${\frac {\pi }{4}}{\frac {r^{4}}{R^{2}}}$. For R the radius of the Earth and r = 1 m, the difference is 0.019 square microns. For r = 1 km, the difference is 190 cm^2.

If you meant one of the other two cases, I'll let you do the math yourself. —Keenan Pepper 03:25, 29 August 2007 (UTC)

I mean the the length of the great circle arc that connects the surface "center" to a point on the circle , the idea is to draw/scratch a circle on the ground with using a stick. Measuring the "radius" using a tape measure. 202.168.50.40 04:50, 29 August 2007 (UTC)

In which case, you can use the formula Keenan gave, but you have to calculate the equivalent "actual radius" for your "great circle arc-length", which is a matter of a little trigonometry. It's a little easier to work in diameters, and then just halve those to get the radii. For the r = 1 m case, the difference is of the order 10^-14 m, so it's not huge, and even at the r = 1 km case it's only 10^-5 m, so a change of less than 2 millionths a of a percent. Basically just remember that the arclength of a circle is the radius times the angle subtended at the centre. Confusing Manifestation 05:31, 29 August 2007 (UTC)

Define a geographic coordinates system such that its pole is a center of your circle (so your circle is a circle of latitude). The meridian arc length from the pole to the equator is now π/2·R. Define φ as the latitude of a point on circle. Let's assume positive φ defines a northern hemisphere, negative the southern one. Now the 'distance' from the equator is d=R·φ. I quoted the word 'distance' because it is positive on the northern hemisphere and negative on the other, while a true distance is an absolute value of that. Now the 'radius' of your circle is r=π/2·R−d=R·(π/2−φ), and its area is S=2πR²(1−sin(φ)).
(Unless I made some mistake...) --CiaPan 05:50, 29 August 2007 (UTC)

Eliminating φ gives an area of 4πR²·sin²(^r/_2R) = 4πr²·(1 − ¹/₁₂ρ² + O(ρ⁴)), where ρ = ^r/_R.  --Lambiam 06:42, 29 August 2007 (UTC)

And if I remember correctly, this tells us that the Gaussian curvature of a sphere of radius R at any point is 1/R². For any Riemannian surface, let A(r,p) be the area of the circle of radius r around the point p (where distance is measured as the original poster intended, along the surface). Then, mimicking Lambiam's formula, A(r,p) = 4πr²·(1 − ¹/₁₂ κ(p)r² + O(r³)), where κ(p) is the Gaussian curvature of the surface at the point p. (Hopefully someone can correct me if there are mistakes in the formula.) Tesseran 07:51, 29 August 2007 (UTC)