1. Conditional Probability
Problem 6.76

2. We choose points at random in a square with sides 0≤x ≤1 and 0≤y ≤1.
Since the area of the square is one, we have a density curve, and the probability that the point falls within any region of the square is the area of that region.
Let X be the x-coordinate and Y be the y-coordinate of any point chosen.

3. Our assignment is to find the conditional probability P(Y<1/2|Y>X).
After drawing the square, next draw the lines y=1/2 and y=x, as we prepare to graph the inequalities.

4. Shade the appropriate portions of the square that meet each condition.
Shade the area where y<1/2.

5. Now shade the area where y>x.
To find the P(Y<1/2|Y>X) we find the area of intersection divided by the area where the condition y>x is met.

6. The area of intersection is a triangle with a base of 1/2 and a height of 1/2 so the area=1/8.
The area where y>x is 1/2.
So,
and the probability is 1/4.