Be very careful when dealing with probability questions, and watch closely if you have the same probabilistic event all the way or you've changed it and need to make corresponding adjustments.
First, I'll show the most general approach, which works very well in this case. Then, I'll analyze yours and give you some advices.
The most general approach when finding probability in geometrical questions is to use definition of probability on plane:
The probability equals to the area of the figure that consists of satisfactory probabilistic events (In our case it consists of all the points within a circle such that x
> 0) divided by the area of the figure that consists of all the possible probabilistic events (In our case we pick a point only within circle so it is a circle itself except circumference).
Let radius of the circle be r
All the possible events:
Area equals πr
Area equals (1/8) × πr
So the probability is [(1/8) × πr
²] / (πr
²) = 1/8.
Your approach is also possible but it is more complicated. let me analyze it:
When you calculate probability in each quadrant separately you change probabilistic event by picking a point not from a circle at whole but just from 1/4 of it that lies in the corresponding quadrant.
So we need to adjust the event by assuming that we pick one of the quadrants first (equiprobably, because the parts of the circle we pick, 1/4 of a circle, all have the same area). And the probability we need to calculate equals to:
(1/4) × P1 + (1/4) × P2 + (1/4) × P3 + (1/4) × P4,
where P1, P2, P3 and P4 probabilities in each quadrant correspondingly.
Let us simplify:
1/4 (P1 + P2 + P3 + P4)
Now, note that x
> y > 0
. So all P2, P3 and P4 are 0. P1, as you've calculated it right, is 1/2. So the answer is:
1/4 (1/2 + 0 + 0 + 0) = 1/8.
Furthermore, if we broke the figure into different areas, like 1/4, 1/8, 1/8, 1/2 of a circle, for example, we would count probability as
(1/4) × P1 + (1/8) × P2 + (1/8) × P3 + (1/2) × P4.