AMS 311 First Test Solutions Spring 2013
7 pt 1. Test A 52x39x26/52x51x50; Test B 4C(13,2)/C(52,2).
8 pt 2. Test A C(9,2)/C(10,3) Test B C(14,3)/C(15,4).
8 pt 3. Test A: Let C = child you first meet is a girl, A1 = GGG, A2 = GGB, A3 = GBG,
A4 = GBB-- P(C) = sum(P(C|Ai)P(Ai) = 1(1/4) + (2/3)(1/4) + (2/3)(1/4) + (1/3)(1/4) =
2/3, Then Find P(A1|C) = P(C|A1)P(A1)/P(C) = 1(1/4)/(2/3) = 3/8;
Test B: Now C = you meet a boy, A1 = BGG, A2 = BGB, A3 = BBG, and A4 = BBB
P(C) = sum(P(C|Ai)P(Ai) = (2/3)(1/4) + (1/3)(1/4) + (1/3)(1/4) + 0(1/4) = 1/3
Find P(A2 or A3|C) = {P(C|A2)P(A2)+ P(C|A3)P(A3)}/P(C) = [(1/3)(1/4) +
(1/3)(1/4)]/(1/3) = (1/6)/(1/3) = 1/2.
6 pt (3,3) 4. Test A a) P(Abar union Bbar) = P(Abar) + P(Bbar) - P(Abar)P(Bbar) = .4 +
.4 - (.4)(.4),
b) P(AB|BCbar) = P(ABBCbar)/P(BCbar) = P(ABCbar)/P(BCbar) =
P(A)P(B)P(Cbar)/P(B)P(Cbar) = P(A) = .6.
Test B a) P(Abar union Cbar) = P(Abar) + P(Cbar) - P(Abar)P(Cbar) = .6 + .6 + (.6)(.6),
b) P(AC|CBbar) = P(ACCBbar)/P(CBbar) = P(ACBbar)/P(CBbar) =
P(A)P(C)P(Bbar)/P(C)P(Bbar) = P(A) = .4.
5 pt 5. Test A: ) 1 - (.75)^7.
Test B: a) sum of C(7,i)(.2)^i(.8)^(7-i), i = 0,1,2,3,
4 pt 6. Test A: C(4,2)(.45)^3(.55)^2.
Test B: C(5,2)(.3)^3(.7)^3.
5 pt 7. Test A: 1 – e^(-4) - .4e^(-4)
Test B: sum e^(-3)3^i/i!, i = 2,3,4
9 pt 8. Test A a) First round win is $6 with prob 1/4, 2nd round win (after loss) is $12-$6
= $6 with prob (3/4)(1/4), 3rd round win (after 2 losses) is $18 - $6 - $12 = 0 with prob
((3/4)^2(1/4), 4th round win (after 3 losses) is $24 - $6 - $12 - $18 = -$12 with prob
(3/4)^3(1/4), and if you lose in all four rounds you have -$6-$12-$18-$24 = -$60 with
prob (3/4)^4.
a) see part b) for prob mass function.
b) E(X) = 6x1/4 + 6x(3/4)(1/4) + 0x(3/4)^2(1/4) + -12x(3/4)^3(1/4) + -60x(3/4)^4
(students do NOT need to evaluate this sum).
Test B: by same reasoning as in Test A (6i replaced by 4i) we have prob. mass func.
and E(X)= E(X) = 4x1/4 + 4x(3/4)(1/4) + 0x(3/4)^2(1/4) + -8x(3/4)^3(1/4) + -40x(3/4)^4
(students do NOT need to evaluate this sum).
8 pt (4,4) 9. Test A: a) 5x4 + 2x19 = 58 (where E(X^2) = Var(X) + [E(X)]^2 = 3 + 16)
b) 5^2Var(X) = 25x3 = 75.
Test B: a) 4x5 + 3x31 = 113, b) 3^2Var(X) = 9x6 = 54 (note: E(X^3) is a constant.