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 xcoordinate 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,
P(Y  | Y  X) 
1
2
and the probability
is 1/4.
1
1
8
2
1

4