Probability question: help

1. Sep 22, 2006

murshid_islam

that day my friend asked me a question. what is the probability that the chord inside a circle (with radius 1) will be greater than D, where D is in the interval [0, 2].

i have came up with the answer
$${\pi - 2\sin^{-1}(D/2)} \over {\pi}$$

is it ok? or did i do anything wrong?

Last edited: Sep 23, 2006
2. Sep 22, 2006

HallsofIvy

Staff Emeritus
Am I missing something? A circle of radius 1 has diameter of length 2 and all other chords are shorter. It is impossible for a chord to have length greater than 2. The probability is 0.

3. Sep 22, 2006

mathman

If you put D=2 in the expression, you will get 0. The question is what is the probability for a specific D, where 0<D<2. The answer will depend on what sort of distribution function defines the placing of the chord - there is more than 1 way to do it.
Examples:
(1) Uniform in distance from the center.
(2) Pick a point on the circumference, other end point is uniform around the circumference.

4. Sep 23, 2006

murshid_islam

how will the 2 distributions affect the answer? can you be a bit more elaborate?

5. Sep 23, 2006

HallsofIvy

Staff Emeritus
Ah, yes, I simply misread the quesition!

6. Sep 23, 2006

mathman

You had to make some sort of assumption about the chords to get the answer you did. I haven't worked out what the rersults would be for the 2 examples I gave, but I can make up possibilities which I know would give different results, although they might look strange. For example, uniform in the square of the distance from the center.

7. Sep 24, 2006

mathman

I took the time to work out the various possibilities I mentioned. To simplify notation, let s=D/2. The probabiliites for these cases are:

Case.......................................................Prob.
uniform in arc length.............................murshid islam result
uniform in distance from center..............(1-s2)1/2
uniform in distance squared...................1-s2
uniform in chord length.........................1-s

As you can see, there is no "right" answer.