## Sunday, November 29, 2009

### Fox 198 This is the beginning of a class of foxes with random processes.
The above process is a special form of a random walk on the plane.
We believe that this process is new! If you see something similar in literature, please let us know.
"Uniformly selected" means that every direction has an equal chance of being selected.
In other words: direction ~
Uniform[0, 360].
Point O is the origin. Generation of point A is irrelevant since OA does not change whether there will be a triangle or not at the end of the process.
Also, it is not important but assume that the random directions are generated clock wise.
The answer has not been confirmed yet, let us know if you find something very different.

1. 2/3*1/6=1/9
I am not sure

I hate probability:)

2. No, 1/9 is not the answer.
THIS FOX HAS NOT BEEN SOLVED YET!

3. I've no idea what "uniform selection" means. Could you pl. explain?

4. A direction is selected by generating a random number between 0 and 360.

Or think it this way:
Point A is a point which is randomly selected on the circle with center O and radius 1. All points on the circle has an equal chance of being selected (Uniform(0,360 degrees)). The process is repeated for point B (now the center is A) and point C (now the center is B).

Let me know if you need more input.

Also check this out:
http://www.mymathforum.com/viewtopic.php?f=13&t=11101

5. The answer is 1/8 because my 10000 trial in excel came close to 0.125. Just kidding. Integral of solution surprisingly came out to 1/8 with all the pi's canceling

6. Actually this can be simulated in Excel. And you should have an answer. I don't think the answer is 1/8. Here is my approach:
a: first angle randomly generated
b: second angle randomly generated

a in [0,60]
Then 0 < b < (180-a)/2 forms a triangle.

a in [60,90]
Then 0 < b < (180-2a) forms a triangle.
Compute probabilities in both case.
Then multiple by 2 (why?)

7. 120

90
__|`-.
__|- `-.
60|______
__|_____|\
__|_____|_\
30|_____|__\
__|_____|___\
__|_____|____\
__0 30 60 90 120 150 180 x

Above area
----------- = 1/12
360*360

Option (C)

8. For a full explanation, with pictures see:
http://home.kpn.nl/henkreuling/solutions8foxes/198_8foxes_solution.pdf

We name α the angle moving from A to B and β the angle moving from B to C with 0º ≤ α, β ≤ 360º, measuring the conventional way (pointing right = angle 0º, pointing left = angle 180º).
It is clear that B is lying on the circle with radius 1 from A.
It is clear that that B must lie on the left half of this circle to make it possible to make a triangle, so it must be 90º < α < 270º.
For α, with 90º < α < 120º the circle with radius 1 and center B intersects OA, in A and another point between O and A. Call this point D. Triangle ABD is isosceles with top-angle 2α – 180º. For angles β with 360º – α < β < (360º – α) + (2α – 180º) = 180º + α point C creates a triangle.
For α, with 120º < α < 180º triangle OAB is isosceles with angle OBC = ½α. For angles β with 180º + ½α < β < (180º + ½α) + ½α = 180º + α point C creates a triangle.
For 180º < α < 270º the situations are (almost) the same because of the symmetry.
Draw a diagram with rectangle [0º,360º]x[0º,360º] with all possible combinations of α and β.
Colour the areas with the above restrictions of the combinations of α and β with C creating a triangle.
With easy calculation you can find that the gray area is 1/12 of the whole area.
9. 