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."

 

[CajunCenturion]

==> What you are saying is that when you measure the water on the two different moist plates AT THE SAME INSTANT there are at least two different correct answers depending upon how you collect/sample the moisture.
This question is about volume and volume requires three dimensions. This is a question about cylinders, not about circles. Further, the original question is about probability distributions which is a completely different exercise.

@Olaf ==> "C didn't say he choose his construction to show, every random point p could be on sme such radial line."
None of the three students explained how the generated their distributions. Why is student C being singled out?

==> C would not need to change his result,
Again, this again shows that you acknowledge that student C's answer is value, since he need not change it. I will grant that you've always acknowledged that his answer was valid.

==> if C was asked, he would certainly say why he choose such a line to represent tho whole sample space, but he didn't.
If C was asked, ... but he didn't. Either C was asked about his methods or he wasn't. How can you possible how he answered a question when the act of asking the question is hypothetical.

It is abundantly clear that default assumptions, whether they be right or wrong, are being used to unfairly discriminate against student C, because his answer, even though correct, is not consistent with those default assumptions. It's also apparent that nothing is going to change that.

--------------
Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
Wise men speak because they have something to say, fools because they have to say something. - Plato

 

[Olaf Doschke]

Well, student A is saying he is considering areas and is correct, student B is stating the same and fails on it, student C is having to explain more on his more complicated thoughts, as you need to comment more complex code, not to make it work correctly, but to show you have thought it through.

That's why he became the main topic here.

Yes, you could ask everyone to better support their solution and A might then still turn out to guess his correct answer. It would also be interesting to see how B thinks in detail, but since noone here got a grip of what he could have been thinking, he was of less interest here.

I strongly disagree, that I discriminate student C, my hypothetical first answer to student C was "good thinking". But he would need to show if this good thinking I notice was really good thinking, was just cunning, was lazyness, or was perhaps provocation. It's not clear he thought that through.

Bye, Olaf.

 

[karluk]

It would also be interesting to see how B thinks in detail, but since noone here got a grip of what he could have been thinking, he was of less interest here.

I don't agree that no one "got a grip of what he could have been thinking". I believe I got a very good grip on his possible thinking, as outlined in my post dated 27 Nov 11 9:39. He could have been modeling a game of darts, where the skill of the dart thrower increases the odds of a hit in the inner circle, beyond what one would expect from a straight calculation of the relative surface areas.

Or take a modified version of SidYuca's proposal to measure the collection rate of water on plates of different areas. Let's say that this puzzle is a model for a small, but intense thunderstorm. The rain falls most heavily near the center of the storm, averaging 1" per hour within a mile of the epicenter. But it tapers off thereafter, averaging only 2/3" per hour between one and two miles from the epicenter. Beyond two miles, the rain tapers off to nothing. If the question is asked, "What fraction of the water falls within a mile of the center of the storm, my calculations indicate that the fraction is 1/3, and B's answer becomes the correct one. B has correctly made his calculations based on the relative areas of the region, and has also avoided A's error in not adjusting for the differing intensity of the rain in different parts of the sample space.

Or B may have simply made an error in calculating the relative areas. Which is true? No one knows without asking B, and SidYuca has prevented us from saying definitively that B's answer is wrong by not taking up my offer of a re-do and clarifying how the point P was selected. Instead, he is hoping that everybody else, including the students and us, are clairvoyant enough to figure out that we are required to assume a uniform distribution based on surface area.

Note that SidYuca made almost an identical error in the statement of the "guess the missing digit" puzzle. The only difference was that in that puzzle we were somehow supposed to be clairvoyant enough to assume a HIGHLY non-uniform distribution of the 882 seven digit candidate numbers. So he reserves the right to mark his students wrong, regardless of whether they assume uniform or non-uniform probability distributions.

 

[CajunCenturion]

==> my hypothetical first answer to student C was "good thinking"
Your hypothetical first answer to student C was "good thinking, but wrong".

--------------
Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
Wise men speak because they have something to say, fools because they have to say something. - Plato

 

[Olaf Doschke]

Yes, and everyone can see that, you don't need to repeat it. Still "good thinking, but wrong" starts with "good thinking", doesn't it?

I later admitted karluk is right and C's answer only is wrong assuming a certain distribution of random points. So what?

Still he didn't made that clear so it's doubtful, he actually has thought about really being right with his answer.

Bye, Olaf.

 

[kwbMitel]

Regarding B's answer. I will apply Occam's Razor and make the least assumptions. The relative size of the White circle that is not covered by black and the back circle is 3:1. Only 1 assumption required. B forgot that Black overlapped White. Only 1 assumption makes this the "Most Likely".

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

 

[SidYuca]

Sid would like to clear up a couple of points for you and himself.

1. He IS the person that made two errors in two posts. He is sorry and will try not to repeat in the future.(For this he writes #2.)

2. He wants to know if the following rewrite would have the same type of response(s).
Note: I am not asking for comments re. student responses but you are free to make them.
-------------------re-write is below-----
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 and their interors, 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 a randomly selected point P, in the interior of the larger circle that it is also in interior of 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."

------------------End of rewrite------

 

[Olaf Doschke]

@kwbMitel:

By seeing the odds of 1:3 it's clear that probabilities must be a multiples of 1/4, as 1+3=4 parts are the whole area, that comes down to 1/4 vs 3/4 and not 1/3 vs 2/3.

See

http://en.wikipedia.org/wiki/Odds

on odds vs probabilities.

So even with your Occam's Razor just making that one assumption you would not come out to 1/3 probability.

@Sid, with your rewritten question I would even more stand with my first answers to each student.

Bye, Olaf.

 

[CajunCenturion]

==> What would you, as a good understanding teacher say to each student, A, B, and C?
In all cases I would say the same thing (which is what I have said many times in these fora).

Ok, you've presented your answer, now provide the math to back it it. Make your case and prove your answer.

Either the proof stands up or it doesn't. Everyone one of those answers can be proven mathematically correct on their own merits. I will grant that some will not accept certain answers because they don't conform to some default assumptions or preconceived expectations, but that's not mathematics.

The math will speak for itself and it will either stand up, or it won't. If it does, the student gets full credit, and if it doesn't, then we still down and go through the math and correct any mistakes.

--------------
Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
Wise men speak because they have something to say, fools because they have to say something. - Plato

 

[SidYuca]

Olaf, I knew I'd get that response from you, but I am still waiting for a rewrite of the question that I will have only the answer 1/4. If you like you may provide 3 rewrites and indicate correct answer for each.

I just want an answer key that is correct for a specific question. It is obvious (to me at least) by virtue of my first try that I can make a question that has 3 different answers.

 

[CajunCenturion]

What is the probability that a randomly selected point P(x, y), where both the x and y Cartesian coordinates are randomly selected directly from a uniform distribution, that lies in the interior of the larger circle also resides in the interior of the smaller circle?

The key point is that if you want a specific answer for a probability question, then you must specify the the domain and its pertinent properties from which the random selection is to be made.

--------------
Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
Wise men speak because they have something to say, fools because they have to say something. - Plato

 

[CajunCenturion]

What is the probability that a randomly selected point P([θ, d), where [θ] and d are randomly selected from uniform distributions of 0 to 2pi, and the radius of the outer circle respectively, lies inside of both circles?

With respect to 1/3, it would take me a bit to determine the correct domain and function, but karluk has described the kind of situation with the thunderstorm that you're trying to model.

All that being said, I would consider turning the question around.

Given two concentric circles of radius 1 and 2, provide the mathematics to show that a randomly selected point P that lies in the outer circle has a
[li]1:4 probability of also being inside the smaller circle,[/li]
[li]1:3 probability of also being inside the smaller circle,[/li]
[li]1:2 probability of also being inside the smaller circle.[/li]


--------------
Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
Wise men speak because they have something to say, fools because they have to say something. - Plato

 

[Olaf Doschke]

I am still waiting for a rewrite of the question that I will have only the answer 1/4

You asked me one via mail and I mailed an answer to you.

Bye, Olaf.

 

[karluk]

If you intend the point P to be selected "uniformly distributed with respect to area", there's no substitute for saying so.

The problem with using phrases like "a randomly selected point P" is that "randomly selected" doesn't automatically imply uniform distribution. Consider the implications of using the phrase in the following sentences:

(1) Take a number selected randomly from {2,3,4,5,6,7,8,9,10,11,12} by tossing a pair of dice.

The numbers are selected randomly but, as is well-known, the randomizing device will favor numbers such as 7 over numbers such as 2 or 12.

(2) Take a number selected randomly from {2,3,4,5,6,7,8,9,10,11,12} by tossing a pair of loaded dice.

In case (1), I believe it is reasonable to assume that fair dice are being tossed. Most people would consider it splitting hairs to the nth degree to quibble with the fact that the word "fair" was left out. In case (2), there obviously is no assumption of fairness at all. We would-be problem solvers await further information to more fully determine how the loading of the dice will affect the distribution of the random numbers being generated.

(3) Take a number selected randomly from {2,3,4,5,6,7,8,9,10,11,12} by spinning a wheel divided into 11 equal sized regions and with each region labeled with one of the numbers.

Here we have abandoned dice as our randomizing device and are using a method that will generate the numbers with equal likelihood. Only in case (3) does "selected randomly" equate with "uniform distribution".

 

[Olaf Doschke]

Thats what my mail to Sid included,

if you just specify the geometry of the random experiment but not how it should be done. I made the example of defining a dice with it's 6 faces and asking what the probility of a 6 is. If you don't define you need to throw the dice in such a way you won't have control over the dice, you can also define you simply put the face with the 6 up always.

I already also here suggested "blindly throwing darts", of course only taken all those into account, that hit the board (larger circle area) at all. Sid's rain example with circles on the ground and assumed the rain distribution is uniform is also ok. You can always argue the laws of physics would prefer a certain non normal distribution of darts or would prefer some number on the upper face of a dice after throwing/rolling it, while seeming random, still follows physical laws. Still the core similar thing of executing the random experiment in such ways is the unpredictable outcome.

"Selecting" a point is a bad term in itself, to select suggests to choose with some preferance, not random selection.

Bye, Olaf.

 

[karluk]

For what it's worth, I have been browsing through one of my old college textbooks and came across a problem that is very similar to this one. It just goes to show that "randomly selected" is a very slippery concept that needs to be dealt with very carefully.

Example 2c. Consider a "random chord" of a circle. What is the probability that the length of the chord will be greater than the side of the equilateral triangle inscribed in that circle?

Solution: The problem as stated is incapable of solution because it is not clear what is meant by a random chord.

The book then works through two plausible ways to generate a "random chord" and comes up with probabilities of first 1/2 and then 1/3 that the length of the chord is greater than the side of an inscribed triangle.

 

[SidYuca]

Olaf
Your statement:
"Selecting" a point is a bad term in itself, to select suggests to choose with some preferance, not random selection."

Indicates to me that you are beatintg a dead horse and did not read the bold in my post.

further your:
"(1) Take a number selected randomly from {2,3,4,5,6,7,8,9,10,11,12} by tossing a pair of dice.

The numbers are selected randomly but, as is well-known, the randomizing device will favor numbers such as 7 over numbers such as 2 or 12."

I speak of making a random i.e unbiased selection. One can make a random unbiased selection from a bag of 10 red and 4 black checkers. Me thinks you are leaning towards those that complained about the 90% problem.

To all the mathematical purists that gave stories about the interior of the circle when I spoke of circles, you take take more effort in attacking the question rather than its solution. I will in the future will not dwell on defense and that may bring out those that stand on the side watching with great wonderment.

I asked for a rewrite of my question with a single correct answer, why is it that nobody will give that to me?

 

[karluk]

I asked for a rewrite of my question with a single correct answer, why is it that nobody will give that to me?

I did suggest a rewrite that would have worked - I suggested that if you meant "uniform distribution with respect to area", that you say so explicitly. There may be other, more subtle, ways to nail down the same concept, but so far your minimalist attempts at problem definition are falling short of eliminating ambiguity. Here is another quote from chapter six of Professor Ross's book that show how a trained mathematician would explicitly state the distribution you seem to prefer:

Example 1d. Consider a circle of radius R and suppose that a point within the circle is randomly chosen in such a manner that all regions within the circle of equal area are equally likely to contain the point. (In other words, the point is uniformly distributed within the circle.)

Professor Ross then poses several questions about this circle, the last of which is

3. Compute the probability that the distance from the origin of the point selected is not greater than a.

Professor Ross then solves a simple double integral and shows that the answer to question 3 is a^2/R^2. This answers your question, since Professor Ross's Example 1d is the same as your circle problem with a=1 and R=2. Plugging these values into Professor Ross's formula, we get 1^2/2^2 = 1/4 and student A's answer is correct, once the problem has been precisely defined.

 

[CajunCenturion]

==> I asked for a rewrite of my question with a single correct answer, why is it that nobody will give that to me?
Please see my posts of 30 Nov 11 9:37 and 30 Nov 11 9:51.

--------------
Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
Wise men speak because they have something to say, fools because they have to say something. - Plato

 

[karluk]

I have been doing a little additional research on this issue, and thought I would pass along what I found, in case anyone wants to pursue the subject. It turns out that the phenomenom of "random selection" frequently being undefined without additional specifics, even if one accepts the the additional assumption that random selection requires uniform distribution, has been known since the late 19th century. It's known as "Bertrand's paradox". The Wikipedia article on the subject is

http://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

Bertrand's original formulation of the paradox involved "randomly selected" chords on a circle. That's undoubtedly the reason for the "random chord" problem I found in my college textbook - Professor Ross was borrowing the example from Bertrand's work. But Olaf and Cajun have shown that the answers of students A and C also represent examples of Bertrand's paradox at work - one gets a different distribution of points depending on whether the "random selection" is on the Cartesian coordinates or polar coordinates of the points in the circle.