CS 201-31, Discrete Structures, Spring 2000, Assignment 9 Solution
Section 3.5, pages 155-156:
16. (using A,B,C,D instead of A1 A2 A3 A4):
|A B C D| = |A| + |B| + |C| + |D|
- |A B| - |A C| - |A D| - |B C| - |B D| - |C D|
+ | A B C| + | A B D| + | A C D| + | B C D|
- | A B C D |
17.
17a). 71 b) 16 c) 8
18.) Let U = {1, 2, 3, ..., 9999}
A = {n | n is divisible by 2}, B = {n | n is divisible by 3}, C = {n | n is divisible by 5}.
Then |A| = 4999, |B| = 3333, |C| = 1999.
A B = {n | n is divisible by 6}, so |A B| = 1666.
A C = {n | n is divisible by 10}, so |A C| = 999.
B C = {n | n is divisible by 15}, so |B C| = 666.
A B C = {n | n is divisible by 30}, so |A B C | = 333.
Using the inclusion-exclusion formula, we see that |A B C| = 7333, so
|(A B C)'| = 9999 - 7333 = 2666.
Section 4.1, pages 185-186:
6: a) = 0.012941 b) = 0.382353
c) 1 - = 0.217376 d) = 0.00995
e) = 0.0000452 f) = 0.000362
7: a) = 0.0072. b) = 0.1476
. c) = 0.3381. d) 1 - (ans. for part b) = 0.8524
e) = 0.5608.
Section 4.2, pages 193-194:
4c) Pr(D E) = 1 - Pr ( (D E ) = 1 - 0 = 1.
Pr (C D) = Pr(C) + Pr(D) - Pr(C D) = Pr(C) + Pr(D) - Pr(C) = Pr(D) = ans to 7d above.
6a) Pr(A B) = Pr(A) + Pr(B) - Pr(A B) = 0.6 + 0.5 - (.6)(.5) = 0.8.
6b) Pr(B C) = Pr(B) + Pr(C) - Pr(B C) = 0.5 + 0.3 - (.5)(.3) = 0.65.
6c) Pr(A B C) = Pr(A)Pr(B)Pr(C) = 0.6(0.5)(0.3) = 0.09.
6d) Pr(A B C) = 1 - Pr(A' B' C') = 1 - (0.4)(0.5)(0.7) = 0.86.
6e) Pr(A' B' C') = (0.4)(0.5)(0.7) = 0.14
Section 4.3, page 202, # 5: A quarter is flipped n=10 times.
a) Pr(exactly 3 heads) = = 120 / 1024 = 0.1172.
b) Pr(exactly 8 heads) = = 45 / 1024 = 0.0439.
c) Pr(at least 8 heads) = = (45 + 10 + 1) / 1024 = 56 / 1024 = 0.0547.
d) Pr(at least 3 heads occur) = 1 - Pr(0, 1, or 2 heads occur)
= 1 - = 1 - (1 + 10 + 45) / 1024
= 1 - 56 / 1024 = 0.9453.