>>13659070We will refer here to the colors of the balls like this:
G;R
Where G is the amount of green balls and R is the amount of red balls.
This signifies we have three boxes: 2;0 , 1;1, and 0;2.
Since we pick at random, we have equal chances of picking either box. Thus , we have 1/3 chance to pick 2;0, 1/3 chance to pick 1;1, and 1/3 chance to pick 0;2.
We pick a green ball: this means this cannot be the 0;2 box, since the number of green balls is 0 (remember, the leftmost number) and thus, we couldn't have taken a green ball out of it.
So, the box we picked is either of these:
the 1;1 box
the 2;0 box.
Now, since we picked a green ball out of either box, the possible boxes become:
1-1;1 = 0;1
2-1;0 = 1;0.
Since our picking chances were equal, and we couldn't have picked the 0;2 box, we're left with equal chances between the two boxes left; so, our chances of being the picked box is of 1/2 for each box (the same).
1;0 : 1/2 chance of being our box
0;1 : 1/2 chance of being our box
Now, the chances of picking a green ball (the leftmost parameter) within each box is:
1;0 : 100% ; 1/2 chance of being box
0;1 : 0% ; 1/2 chance of being box
Thus, in 50% of cases, we have a 100% chance of getting a green ball, and in 50% of cases, we have a 0% chance of getting a green ball.
Thus, we will get a green ball 1/2 of the time.
QED.
