Problem 2 Let (2,,P) be a probability space. (1) For measurable events A,B,CE A with P(C) >0 and P(BnC)>0, show that P(AnBnC)=P(A BOC)P(B|C)P(C). (2) Let A be a measurable event and let B₁,...,B, EA as well as C₁,...,Cm E be partitions of the sample space 2, i.e. BnB, =ø for i #j; CnCe = for k #l; and Show that m n=ÜB₁ =ÜC₁. j=1 n m P(A)=P(A|B₂NC,)P(B, C,)P(C₂). i=1j=1 if P(C) >0 and P(B; nCj) >0 for all i = 1,...,n and j = 1,..., m. (3) Let X, Y, Z be random variables on (2, 4, P) with possible values x₁,...,xe € R for X; y1,...,ym € R for Y; and 21,...,Zn ER for Z. Suppose that P(Z = z) >0 and P(Y = yj, Z = zk) > 0 for all k = 1,...,n and j = 1,...,m. Show that for every i = 1,...,l m n P{X = x₁}= [[P{X = x₁ | Y = yj, Z = zk} P {Y=yj | Z = zk} P {Z = zk}. j=1k=1

A 3rd year student has 75 credits and a cumulative GPA of 2.88. This semester she is registered for 15 credits and hopes to raise his cumulative GPA to 3.00 and make the Dean’s List.   Is it possible? What semester GPA will she need to accomplish that? Show your calculations or post a screenshot from your online GPA grade calculator.

ignore the question above please just do #7

Please do #20 as well