Answer:
To determine the conditional probability that the x coordinate of the point exceeds its y coordinate, given that the x coordinate of the point exceeds 0.5, we need to consider the area of the unit square where the x coordinate is greater than 0.5 and the y coordinate is less than the x coordinate.
First, let's visualize the unit square: {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.
The unit square is a square with side length 1, centered at the origin (0,0), and bounded by the points (0,0), (1,0), (0,1), and (1,1).
Next, we need to identify the area within the unit square where the x coordinate is greater than 0.5. This is the right half of the unit square, from x = 0.5 to x = 1.
Now, we need to determine the area within this region where the y coordinate is less than the x coordinate. This is a triangular region within the right half of the unit square, where the base of the triangle is the line y = x (since the x coordinate exceeds the y coordinate) and the height of the triangle is the line y = 0.5 (since the x coordinate exceeds 0.5).
The area of this triangular region can be calculated as half the product of the base and height. In this case, the base is (1 - 0.5) = 0.5 and the height is 0.5. So, the area of the triangular region is (1/2) * 0.5 * 0.5 = 0.125.
Finally, to calculate the conditional probability, we divide the area of the triangular region by the area of the right half of the unit square. The area of the right half of the unit square is 0.5.
Therefore, the conditional probability that the x coordinate exceeds the y coordinate, given that the x coordinate exceeds 0.5, is:
0.125 / 0.5 = 0.25.
So, the conditional probability is 0.25.
Explanation: