Conditional Probability Problem 6.76

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.

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.

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

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.

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.