4 Conditional Probability Mass Function
Assume that a random variable x is Bernoulli(p) with
P[x]=[x]={(p,x=1),(1-p,x=0.):}
If x=1, a point is selected at random from region A, and if x=0, a point is selected at random from region B, where A and B are as shown in Figure 3. If the point is selected in an upper quadrant of the plane uv, we set Y=1, and if a point is selected in a lower quadrant of the plane uv, we set Y=0.
The sample space in this problem is formed by the regions A and B, and the probability of selecting a point in a specific quadrant in A or B is directly proportional to the area of that quadrant (with respect to the area of A or B ).
(a) Find the conditional probabilities P[Y]=y|x|=[x] for x=0,1 and y=0,1.
(b) Find the probability of Y=0. That is, find P[Y]=[0].
Note: use the law of total probability.
BO3423aC2d23A
B
A
1 a
+
2
1 C
d
3
2
3