Question

Three numbers (a, b, and c) are picked at random in the range of [-5,5]. Consider events X = {a > 1, b > 1}, Y = {-2 < a, b, c < 3}, and Z = {c >= 0}. Determine the following:

a) P(X), P(Y), and P(Z)

b) P(X|Y), P(Y|Z), and P(Z|X)

c) Whether any pairs of the above events are independent

Answer 1

a)P(X)=P(a>1)*P(b>1)*P(c)=(4/10)*(4/10)*(10/10)=0.16

P(Y)=P(-2<a<3)*P(-2<b<3)*P(-2<c<3)=(5/10)*(5/10)*(5/10)=125/1000=0.125

P(Z)=P(a)*P(b)*P(c)=(10/10)*(10/10)*(5/10)=0.5

b)P(X|Y)=P(X n Y)/P(Y)=P(1<a<3)*P(1<a<3)*P(-5<a<3)/P(Y)

=(2/10)*(2/10)*(5/10)/(0.125)=20/125=0.16

P(Y|Z)=P(-2<a<3)*P(-2<b<3)*P(0<c<3)/0.5 =(5/10)*(5/10)*(3/10)/0.5=0.15

P(Z|X)=P(a>1)*P(b>1)*P(c>0)/0.16=(4/10)*(4/10)*(5/10)/0.16=80/160=0.5

c)

as P(X|Y)=P(X) as well P(Z|X)=P(Z) ; therefore X and Y as well Z and X are independent