BULLS' EYE- Students A, B & C 5

[SidYuca]

This happened in a class. What would you, as a good understanding teacher say to each student, A, B, and C?


Consider two concentric circles with center at O, where the radius of the smaller circle (interior is Black) = 1 and the first drawn larger circle with white interior having radius = 2.

What is the probability that if a point, P, is selected in the larger circle that it is also in the smaller one?

ANSWERS
Student A: "1/4 considering the areas my answer appears correct."

Student B: "1/3 Considering areas my answer appears correct."

Student C: "1/2 Consider drawing the radius OB of the larger circle which passes through point P and intersects the smaller circle at A. Since OA = AB and P is on one of them, my answer appears correct."

 

[IPGuru]

To all three I would say

recheck your work & come back when you KNOW you have the correct answer & show why, not just one that appears correct. I do not accept Guesses

But I don't claim to be an understanding teacher, I like to make students work

I think I know the correct answer but I am now looking to see if I have a flaw in my reasoning.
[hide]
I go with answer A
[/hide]

Computers are like Air conditioners:-
Both stop working when you open Windows

 

[karluk]

Spoiler

As the problem is given, the probability is undefined. The teacher has neglected to specify a probability density function to determine how the point P is selected. Once the probability density function has been stated, the probability of P being in the inner circle could turn out to be anywhere between 0% and 100%, inclusive. So the teacher should tell all three students, "Well done! Give me a probability density function that leads to the probability you've chosen, and I'll give you full credit for your work."

This problem is highly reminiscent of the seven digit number puzzle from an earlier thread. SidYucca apparently intended to present that problem with a distribution function that resulted in generating seven digit numbers that were unusually rich in multiples of nine. But he failed to define the problem tightly enough and allowed a smart-aleck student named "karluk" to pick a different distribution that led to some unintended results.

 

[Olaf Doschke]

[hide]
Student A: correct guess, but you owe a more detailed explaination.

Student B: good thinking, but wrong. The odds are 1:3 and the probabilites have the same relation to each other, but that means 1/4 vs 3/4, not 1/3 vs 2/3, which would translate to odds of 1:2.

Student C: good thinking, but wrong. You're thinking in polar cordinates of distance to O and angle, eleminating the angle as unimportant for the problem, as the inner points merely differ from the outer points in the distance from 0, not in the angle.

But the angle is still important, the polar coordinate system has a spatial uneven nature, with higher distance, same angle delta mean a longer arc, you need to consider that.
[/hide]

Bye, Olaf.

 

[IPGuru]

I once had a teacher who did almost exactly what I described I when presented with an answer would ask the student are you sure?
the student would go back, check the work & finally say the could not see anything wrong & ask what they had missed.

the teacher would then say I didn't say there was anything wrong just asked if you were sure about your answer.

Mean but effective.

Computers are like Air conditioners:-
Both stop working when you open Windows

 

[kwbMitel]

[hide] I think can see the logic of each answer but I have a preference.

A - Considers the area of The larger circle with respect to the smaller. 3/4 of the White circle does not contain the black circle. Thus 1/4 of the time point P will be within the Black circle. This would be my preference

B - This one is more difficult for me and might be wrong. Similar to above but considers the shared space differently. 1/4 and 1/4 cancel out? leaving 1/3rd? I can't find a reasonable mathematical reason for this to be true.

C - Is converting a 3 dimentional problem into a 2 dimentional problem. Is ignoring a lot of other space but does not appear to be invalid. Student is defining how he wants to solve the problem and then solving it that way. There could be hundreds of other solutions this way with different results. Example, the circles could be drawn on a sphere introducing other variables not mentioned (or prevented) in the original problem.

I would answer A normally
I would answer C,D,E,F etc if I felt like a smartass
I would not answer B as I cannot fully see the logic.
[/hide]

**********************************************
What's most important is that you realise ... There is no spoon.

 

[kwbMitel]

Just read the other answers

[hide] Edit my answer and replace smartass with Karluk. [/hide]



**********************************************
What's most important is that you realise ... There is no spoon.

 

[SidYuca]

karluk

Hoisted/foiled by my own petard! I was thinking of you when I was putting this thing together. The original version had a dart board but I didn't want to explain away missing the dart board, then I thought of putting the circles flat on the bottom of a 4 unit diameter well with no air currents or temerature gradients etc. In my my fervent desire to not hear from you again as in the past and get humbled again you have struck again. Your sharp foil has struck me deep once again. But I am not disappointed by your thrusts I have read all the past puzzles and have read your comments and labeled your reputation. People of your specific talents were always welcome in problem solving meetings I was involved with 20 years ago (before my retirement)to help define and clarify the problem which they always did at any expense. They provided a valuable service at the beginning of each session. Sometimes they continued to contribute to the solution (sometimes not). Sometimes they unilaterally rewrote the problem which did cause some concern.
Please consider solving this problem after inserting my missing "random".

P.S. I think I gave you a star or something but not sure if it went through.

 

[Olaf Doschke]

Sid,

I thought besides the math problem you were asking for a teacher's ability to spot how students are thinking to get to their results and to put their thinking on the right track with the responses to them.

But kudos to karluk, you can of course also bend the problem to match the solution. Not unusual in IT business.

Bye, Olaf.

 

[Olaf Doschke]

That aspect also reminds me of this:

http://1.bp.blogspot.com/_ovfHMsXrVow/TEet1buu_TI/AAAAAAAAADQ/ZOrYH6zVbxk/s1600/design-process.jpg

Bye, Olaf.

 

[SidYuca]

Olaf said


"I thought besides the math problem you were asking for a teacher's ability to spot how students are thinking to get to their results and to put their thinking on the right track with the responses to them."

You are exactly correct.
I would also like to see responses to responses.

re. your response to C. I don't know where polar coordinates comes in though. And I don't understand your reference to the angle being a factor in C's thought process. It is the point P (anywhere in the sample space except at O) which defines the radius that Student C references. Note: If P is the center then any radius will serve.

I don't get the thought behind this being a 3D problem.

 

[kwbMitel]

@ Sid [Hide]Sorry, I realized my error earlier re:3d, I got ahead of myself as 3d is another smartass(Karluk) style answer that would yield a different result than the 3 already provided. C was ignoring the relative areas (infinite angles) of the circle and focusing on the one angle that mattered to him. Not invalid as long as he defines how. (as he did). 3D comes into play if the circles are drawn on a Sphere that has a radius greater than 2. Don't even get me started on irregular shapes of 3D objects or perspective, I'll drive you crazy[/hide]

**********************************************
What's most important is that you realise ... There is no spoon.

 

[kwbMitel]

Oh, and Sid. I you have not yet found the STC forum check it out as puzzles such as this one aren't really for programmers.

forum1229

**********************************************
What's most important is that you realise ... There is no spoon.

 

[Olaf Doschke]

2D spatial

[hide]
Well, as you say it C is talking of the radius, in polar coordinates that is one part and the other coordinate is the angle.

If you would inverse the problem and say you want to write a program to create random points P you would perhaps start with x,y each in the range of [-2,2] and checking if the point is within the outer and then inner circle, but that makes you loose time for all the points outside. Transform the problem into polar coordinates and the generation of coordinates inside the outer circle is much easier, you create two random numbers radius [0,2] and an angle [0,360[.

That would lead to a distribution of points in the sense of karluk. As the two random numbers are independant and evenly distributed you would get half of those random points inside and half of them outside of the inner circle, because the angle doesn't matter and the evenly distribution of radius would lead to a 1:1 insead of 1:3 distribution of points.

So your problem also has a meaning in generating pseudo random numbers or points, done the wrong way you get a non normal distribution, if you are like most of us, seeing A as the right solution.

The point is: The radius defines the density of points, if you plot a circle with N points, their distance get's higher, the higher the radius is, and therefore a real random distribution of points - as in blindly throwing darts - is not generated by determining random radius and angle, but random x and y.

Write a program determining random points in both ways and see how the distribution of points will differ. It should be significantly visible how uneven the random polar coordinates will distribute generated that way.
[/hide]

Bye, Olaf.

 

[Olaf Doschke]

Oh and,

[hide]
The angle is indeed eleminated in C's tought process, and that's wrong.
[/hide]

Bye, Olaf.

 

[SidYuca]

Olaf,
I am not sure if there is or isn't a 1 to 1 mapping between points in the inner circle and the annulus (I need to study that), but I do know there is a 1 to 1 mapping between the points on the base of a triangle and the segment determined by midpoints on the other two sides (a tease for you) and that the density Question has little or nothing to do with C's response.

I have no doubt that A has the correct answer and fairly sure by his mention "considering the areas" that he 'knows' and no questioning about such things of how, his knowledge of formulas, value/definition of Pi has little to be gained. This is not to say that he didn't cheat!

That being said it appears to me that B needs some further 'talking to/education'. I await the more responses from the group.

As for C, he may be the 'smartest' of the group. His answer is not correct but it is not an uncommon error made by many experimentors and Phd candidates. And I await more comments on his 'experimental' error.

As for Circle Squared, I was a member in it's earliest stages and was 'univited' by its 'leader'. I ocassionally lurk there but not more and that's enough of this.

 

[Olaf Doschke]

Sid,

simply consider generating the random points with radius and angle as in:

  angle = Random()*2*pi
  radius = Random()*2

With Random() being a random number in the range [0..1[ and Random() itself being equal distributed. Then about half of that points have a radius<=1, the angle of course does not matter in respect of inner circle vs annulus.

I made a picture of that and you can see that the distribution is unevenly with a higher weight on inner points than you would consider random:

So while you the coordinates are valid and random, they are not evenly distributed and so it would be "unfair" but would lead to the 1:1 distribution.

You don't need to make the experiment, if you take it for granted random() does generate as many values below .5 than above .5 in the long run. That means the points will be inside and outside the inner circle in about the same amount in the long run.

The distribution looks like this:

http://dl.dropbox.com/u/17129485/points.png

On the left side the points are generated that way, on the right side via random x,y valus, only drawing those points inside the larger circle. I didn't draw the inner r=1 radius, but you get the picture.

Answer C can said to be right in the sense of karluk with that kind of generating the points, as that distribution of points is having a higher densitiy towards the center.

Bye, Olaf.

 

[SidYuca]

Olaf
You missed my point (or was it a hook) when I said:

"I am not sure if there is or isn't a 1 to 1 mapping between points in the inner circle and the annulus (I need to study that), but I do know there is a 1 to 1 mapping between the points on the base of a triangle and the segment determined by midpoints on the other two sides (a tease for you) and that the density Question has little or nothing to do with C's response."

I still am not sure of the first half of my statement above even after seeing your picture and what looked like a darker center of one circle. But darkness should not be confused with lack of unpainted points. I grant you that these 2 methods may produce different 'designs' but back to the point in the problem of one area having more points than the other.

I encourage you to look at my wording of the 'tease' above and tell me your opinion re. my statement of shorter segment vrs longer segment.

 

[CajunCenturion]

Just a quick reminder on the difference between a normal distribution and a uniform distribution so that we use the proper terminology.

[COLOR=white white]A normal distribution is shaped as a bell curve where just over 68% of the data points lie within one standard deviation of the mean and about 95% lie within two standard deviations of the mean.

A uniform distribution is the one where all data points have an equal probability of occurrence.  This is the type of distribution created by almost all program psuedo random-number generator functions.  Answer "A" presumes a uniform distribution of point selection.[/color]

==> I am not sure if there is or isn't a 1 to 1 mapping between points in the inner circle and the annulus
Perhaps we should have a puzzle based on degrees of infinity.

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[Olaf Doschke]

Thanks Cajun,

So to make myself clearer I'm talking of a uniform distribution of random values generated by the Random() function. And thus it's easy to see even without studying it in detail, that this kind of random point generation is generating as much points with radius<1 than with <=radius<2 using the formula radius = Random()*2.

And I disagree that "the density Question has little or nothing to do with C's response."

Simple reasoning: C is eleminting the angle from the problem, assuming a uniform distribution of radiusses alone, but a uniform distribution of points in the circle will not lead to such a uniform radius distribution. In fact the distribution of the radiuses of uniformly distributed random points would be radius = Sqrt(Random())*2. Sqrt() being the square root. I teasy you to verify that.

The thought of eleminating the angle due to geometric symmetry is wrong, student C is indirectly introducing a nonuniform distribution with that assumption. For a uniform distribution of points you will need a number of points proportional to the circumference of the radius, which yields morepoints with higher radius than with shorter.

I actually don't understand your other geometry puzzle "there is 1 to 1 mapping between the points on the base of a triangle and the segment determined by midpoints on the other two sides".

In what way do two midpoints determine a segment? At first glance they only define a line. And in what way do you talk of a mapping towards base line points?

Bye, Olaf.