Mathematics 20
Spring 2003
Syllabus
326  1.2
Discrete probability distributions 
Hand in page 35: 2, 3, 4, 6, 22. Do (don't hand in): 1, 5. 
328  1.2 continued, 3.1 Counting  Hand
in page 35: 9, 11ac, 13, 18b. Do(don't hand in): 25, 31a Hand in page 88: 2, 12 Do(don't hand in): 1, 3, 5 
331 
3.1, 3.2 Counting, permutations,
combinations 
Hand in page 88: 7; page 113:
1a,c,d,f,g, 2. Also hand in: A
permutation of the set {1, 2, 3} is chosen at random. What
is the probability that it does not have a fixed
point? Do on page 88: 17 and on page 113: 13. 
42 
More combinations, Pascal's triangle 
Hand in 3.1: 14; 3.2: 9, 20, 22.
Do 3.1: 13; 3.2: 19 
44 
Bernoulli trials 
Hand in 3.2: 1b,e, 6, 8, 10
Do 3.2: 7, 15. In problem 10, use the program
BinomialProbabilities available in Chapter 3 here.
(If necessary, read the instructions at the
bottom of the page the link takes you to.) 
47 
Inclusionexclusion
principle 
Hand
in 3.2: 18, 34(a) and these two
problems. Some solutions

49 
Conditional
probablities 
Hand
in 4.1: 1, 2, 5(a), 8, 24. Do 4.1:
3, 7, 9, 29 
411 
Tree
diagrams 
Hand
in 4.1: 12, 16, 17(a), 19, 22. Do: 13, 15,
23 (Just find a way to distrubute the balls so that P(a
white ball) is > 1/2.) 
414 
Independent
random variables, expected value 
Hand
in 4.1: 35, 39, and 6.1: 2, 4, 8. Do 6.1: 3, 5

416 
More
expected values

Hand
in p. 36: 14, and 6.1: 14, 16(for part (b), just compute
P(total = 21), 18, 20ab, 36. Do 6.1: 17 
418 
Variance 
Hand
in 6.1: 22(You'll need to use the BinomialProbabilities
program for this) and 6.2: 2, 4, 12, 14. Do 6.2: 3 
421 
More variance 
Hand in 6.2:
8, 10, 20
Do: 7, 11, 15 
423 
Hand in 6.2:
27 and 5.1: 6, 7, 8 and problem 3 here.
Do 5.1: 1 

425 
Poisson distribution 
Hand in 5.1:
12, 14(a), 16, 22, 18, 20. Do 5.1: 13,
17, 21, 25, 27. Use the Poisson distribution on all of these,
even if the instuctions don't explicitly say to approximate probablitities. 
428 
More Poisson,
start convolutions 
Hand in 5.1:
14(b); 6.1: 21; and problem 4 here; Do 5.1: 23 
430 
Convolusions,
Chebyshev's inequality 
Hand
in 5.1: 32; 7.1: 2, 4(Just say what the probability
is that you win 20 after 10 independent plays, and say which, among
all possible winnings, is the most probable. How is this related
to your expected winnings?); 8.1: 1, 5. Do: 7.1: 3 
52 
Law of Averages 
Hand in 8.1:
6, 8 and problem 5 here.
Do 8.1: 7, 9, 11 
55 
Central limit
theorem 
Hand in 9.1:
1ab, 2ab, 3
Do 9.1: 1cd, 2c, 5, 9 
57 
More central
limit theorem 
Hand in 9.1:
4(just for 499,000 and 501,000), 8, 12; 9.2: 2, 6; and problem
6 here. Do 9.1: 13 and 9.2: 1, 5, 7 
59 
CLT and statistics 
Hand in 9.1:
14, 16 Do: 17, 18 
512 
test 

514 
Start Markov
chains 
Hand in 11.1:
2, 4, 6(w should be u, as in Theorem 11.2 on page 407), 8, 12
Do 11.1: 3, 5, 7, 11 
516 
Absorbing chains 
Hand in 11.2:
2, 4, 18ab and problem 7 here.
Do: 1, 3, 5(first part) 
519 
More absorbing
chains 
Hand in 11.2:
6, 8, 9 Do 11.2: 5, 15, 18c(answer
is .67) 
521 
Regular Markov
chains 
Hand in 11.3:
2ab, 10(11.10 in on p. 411) and
Problem 8 here. Do
11.3: 1, 3, 11 
523 
Ergodic Markov
chains 
Hand in
11.3: 2c, 4, 24 and 11.5: 2b, 6 (Due Wed., the
28th, but if it's in the box by noon Monday, the 26th, it will be graded by
Wed.) Do 11.3: 5, 13 and 11.5: 3, 7 
528 
Mean recurrence times 
Suggested 11.5: 2a(answer: 4.5),
4(answer:1+a/b for yes, 1+b/a for no), 5 