Probability of a random point in a disk? Let a point be picked at random in the disk of radius $1$. Find the probability that it lies in the angular sector from 0 to $\pi$/$4$ radians. 
I just started teaching myself probabilty and came to this question. I found this in an Intro to Probability text and was wondering how would we solve this type of problem.
If I had to guess, my answer would be $1/8$ or $\pi$/$8$. 
Here is the work:
$\pi$/4 $*$ $1/2$ since the disk is a radius of $1$.
Is this correct? If so, can you please show if there another way that I can solve this problem. 
 A: The answer is indeed $1/8$, and here's a justification: 
We're choosing a point on the disk at random, which means here that each point on the disk is equally likely to be chosen. It follows that the probability of choosing a given region is proportional to its area.
The sector from $0$ to $\pi/4$ takes up $\frac18$ of the area of the circle.  Since any $\frac18$ of the circle has the same chance of being chosen, the probability that our sector is selected is $1/8$, since we are choosing $1$ eighth of the disk out of $8$ equally likely possibilities.
A: The desired probability is the ratio of the area of this angular sector to the area of the entire disk. Since area is proportional to the size of the central angle, it follows that the desired probability is:
$$
\dfrac{\pi/4}{2\pi} = \dfrac{1/4}{2} = \dfrac{1}{8}
$$
A: By symmetry, there is nothing to choose between the eight octants; so the probability is equal for each of them: one eighth.
A: Your answer is fine and it is independent of the radius:
$$
\int_{0}^{a}{2r \over a^{2}}\,{\rm d}r
\int_{o}^{\pi/4}\,{{\rm d}\theta \over 2\pi}
=
{1 \over 8}
$$
A: there is a chance of 1 to put it anywhere on the disk. (ian assumption i make but i guess it is a good one)
the area of the disk is ?1
the area that counts as a win is ?2
the area of ?2 / ? 1 is your probability.
Probabilites are always relative and are always between 0 and 1 (including both) so if you get ananswer that is bigger than one, or can change the question so the answer gets bigger than 1 something is wrong   
