n = 10000; % Number of draws.
X = ceil(52*rand(1,n));
aces = (1 <= X & X <= 4);
naces = sum(aces);
fprintf(’There were %g aces in %g draws.\n’,naces,n)
In Example 1.12, we showed that the probability of drawing an ace is 1/13 ≈ 0.0769.
Hence, if we repeat the experiment of drawing a card 10 000 times, we expect to see about 769 aces. What do you get when you run the script? Modify the script to simulate the drawing of a face card. Since the probability of drawing a face card is 0.2308, in 10000 draws, you should expect about 2308 face cards. What do you get when you run the script?
26. A new baby wakes up exactly once every night. The time at which the baby wakes up occurs at random between 9 pm and 7 am. If the parents go to sleep at 11 pm, what is the probability that the parents are not awakened by the baby before they would normally get up at 7 am? Specify your sample spaceΩ and probability P.
27. For any real or complex number z = 1 and any positive integer N, derive the geomet-ric series formula
N−1 k=0
∑
zk = 1− zN
1− z , z = 1.
Hint: Let SN:= 1 + z + ··· + zN−1, and show that SN− zSN = 1 − zN. Then solve for SN.
Remark. If |z| < 1, |z|N→ 0 as N → ∞. Hence,
∑
∞ k=0zk = 1
1− z, for |z| < 1.
28. Let Ω := {1,...,6}. If p(ω) = 2 p(ω− 1) forω= 2,...,6, and if ∑6ω=1p(ω) = 1, show that p(ω) = 2ω−1/63. Hint: Use Problem 27.
1.4: Axioms and properties of probability
29. Let A and B be events for which P(A), P(B), and P(A ∪ B) are known. Express the following in terms of these probabilities:
(a) P(A ∩ B).
(b) P(A ∩ Bc).
(c) P(B ∪ (A ∩ Bc)).
(d) P(Ac∩ Bc).
30. Let Ω be a sample space equipped with two probability measures, P1andP2. Given any 0≤λ≤ 1, show that if P(A) :=λP1(A)+(1−λ)P2(A), then P satisfies the four axioms of a probability measure.
31. Let Ω be a sample space, and fix any pointω0∈ Ω. For any event A, put µ(A) :=
1, ω0∈ A, 0, otherwise.
Show thatµsatisfies the axioms of a probability measure.
32. Suppose that instead of axiom (iii) of Section 1.4, we assume only that for any two disjoint events A and B,P(A∪B) = P(A)+P(B). Use this assumption and inductionj on N to show that for any finite sequence of pairwise disjoint events A1,...,AN,
P
N
n=1
An
=
∑
Nn=1P(An).
Using this result for finite N, it is not possible to derive axiom (iii), which is the assumption needed to derive the limit results of Section 1.4.
33. The purpose of this problem is to show that any countable union can be written as a union of pairwise disjoint sets. Given any sequence of sets Fn, define a new sequence by A1:= F1, and
An := Fn∩ Fn−1c ∩ ··· ∩ F1c, n ≥ 2.
Note that the Anare pairwise disjoint. For finite N≥ 1, show that
N n=1
Fn =
N n=1
An.
Also show that
∞ n=1
Fn = ∞
n=1
An.
jIn this case, using induction on N means that you first verify the desired result for N= 2. Second, you assume the result is true for some arbitrary N≥ 2 and then prove the desired result is true for N + 1.
34. Use the preceding problem to show that for any sequence of events Fn,
35. Use the preceding problem to show that for any sequence of events Gn,
P
36. The finite union bound.Show that for any finite sequence of events F1,...,FN,
P
Hint: Use the inclusion–exclusion formula (1.12) and induction on N. See the last footnote for information on induction.
37. The infinite union bound.Show that for any infinite sequence of events Fn, P
Hint: Combine Problems 34 and 36.
38. First Borel–Cantelli lemma.Show that if Bnis a sequence of events for which
∑
∞ of the preceding problem, and the fact that (1.36) impliesN→∞lim
∑
∞n=NP(Bn) = 0.
39. Let Ω = [0,1], and for A ⊂ Ω, put P(A) := A1 dω. In particular, this impliesP([a,b])
= b−a. Consider the following sequence of sets. Put A0:= Ω = [0,1]. Define A1⊂ A0
by removing the middle third from A0. In other words, A1 = [0,1/3] ∪ [2/3,1].
Now define A2⊂ A1by removing the middle third of each of the intervals making up A1. An easy way to do this is to first rewrite
A1 = [0,3/9] ∪ [6/9,9/9].
Then
A2 =
[0,1/9] ∪ [2/9,3/9]
∪
[6/9,7/9] ∪ [8/9,9/9] .
Similarly, define A3by removing the middle third from each of the above four inter-vals. Thus,
A3 := [0,1/27] ∪ [2/27,3/27]
∪ [6/27,7/27] ∪ [8/27,9/27]
∪ [18/27,19/27] ∪ [20/27,21/27]
∪ [24/27,25/27] ∪ [26/27,27/27].
Continuing in this way, we can define A4, A5, . . . . (a) ComputeP(A0), P(A1), P(A2), and P(A3).
(b) What is the general formula forP(An)?
(c) The Cantor set is defined by A :=∞n=0An. FindP(A). Justify your answer.
40. This problem assumes you have read Note 1. Let A1,...,Anbe a partition ofΩ. If C := {A1,...,An}, show thatσ(C ) consists of the empty set along with all unions of
the form
i
Aki
where kiis a finite subsequence of distinct elements from{1,...,n}.
41. This problem assumes you have read Note 1. Let Ω := [0,1), and for n = 1,2,..., letCndenote the partition
Cn :=k− 1 2n , k
2n
,k = 1,...,2n .
LetAn:=σ(Cn), and put A :=∞n=1An. Determine whether or notA is aσ-field.
42. This problem assumes you have read Note 1. Let Ω be a sample space, and let X:Ω → IR, where IR denotes the set of real numbers. Suppose the mapping X takes finitely many distinct values x1,...,xn. Find the smallestσ-fieldA of subsets of Ω such that for all B⊂ IR, X−1(B) ∈ A . Hint: Problems 14 and 15.
43. This problem assumes you have read Note 1. Let Ω := {1,2,3,4,5}, and put A :=
{1,2,3} and B := {3,4,5}. Put P(A) := 5/8 and P(B) := 7/8.
(a) FindF :=σ({A,B}), the smallestσ-field containing the sets A and B.
(b) ComputeP(F) for all F ∈ F . (c) Trick question. What isP({1})?
44. This problem assumes you have read Note 1. Show that aσ-field cannot be count-ably infinite; i.e., show that if aσ-field contains an infinite number of sets, then it contains an uncountable number of sets.
45. This problem assumes you have read Note 1.
(a) LetAαbe any indexed collection ofσ-fields. Show thatαAαis also aσ-field.
(b) Illustrate part (a) as follows. LetΩ := {1,2,3,4},
A1:=σ({1},{2},{3,4}) and A2:=σ({1},{3},{2,4}).
FindA1∩ A2.
(c) LetC be any collection of subsets of Ω, and letσ(C ) denote the smallestσ -field containingC . Show that
σ(C ) =
A :C ⊂A
A ,
where the intersection is over allσ-fieldsA that contain C .
46. This problem assumes you have read Note 1. Let Ω be a nonempty set, and let F andG beσ-fields. IsF ∪ G aσ-field? If “yes,” prove it. If “no,” give a counterex-ample.
47. This problem assumes you have read Note 1. Let Ω denote the positive integers.
LetA denote the collection of all subsets A such that either A is finite or Acis finite.
(a) Let E denote the positive integers that are even. Does E belong toA ?
(b) Show thatA is closed under finite unions. In other words, if A1,...,Anare in A , show thatni=1Aiis also inA .
(c) Determine whether or notA is aσ-field.
48. This problem assumes you have read Note 1. Let Ω be an uncountable set. Let A denote the collection of all subsets A such that either A is countable or Acis countable.
Determine whether or notA is aσ-field.
49. The Borelσσ-field.This problem assumes you have read Note 1. LetB denote the smallestσ-field containing all the open subsets of IR := (−∞,∞). This collection B is called the Borelσσ-field. The sets inB are called Borel sets. Hence, every open set, and every open interval, is a Borel set.
(a) Show that every interval of the form(a,b] is also a Borel set. Hint: Write (a,b]
as a countable intersection of open intervals and use the properties of aσ-field.
(b) Show that every singleton set{a} is a Borel set.
(c) Let a1,a2,... be distinct real numbers. Put
A := ∞
k=1
{ak}.
Determine whether or not A is a Borel set.
(d) Lebesgue measureλ on the Borel subsets of (0,1) is a probability measure that is completely characterized by the property that the Lebesgue measure of an open interval(a,b) ⊂ (0,1) is its length; i.e.,λ(a,b)
= b − a. Show that λ(a,b]
is also equal to b− a. Findλ({a}) for any singleton set. If the set A in part (c) is a Borel set, computeλ(A).
Remark. Note 5 in Chapter 5 contains more details on the construction of prob-ability measures on the Borel subsets of IR.
50. The Borelσσ-field, continued.This problem assumes you have read Note 1.
Background: Recall that a set U⊂ IR is open if for every x ∈ U, there is a positive numberεx, depending on x, such that(x −εx,x +εx) ⊂ U. Hence, an open set U can always be written in the form
U =
x∈U
(x −εx,x +εx).
Now observe that if(x −εx,x +εx) ⊂ U, we can find a rational number qxclose to x and a rational numberρx<εxsuch that
x∈ (qx−ρx,qx+ρx) ⊂ (x −εx,x +εx) ⊂ U.
Thus, every open set can be written in the form
U =
x∈U
(qx−ρx,qx+ρx),
where each qxand eachρxis a rational number. Since the rational numbers form a countable set, there are only countably many such intervals with rational centers and rational lengths; hence, the union is really a countable one.
Problem: Show that the smallestσ-field containing all the open intervals is equal to the Borelσ-field defined in Problem 49.
1.5: Conditional probability
51. MATLAB. Save the following MATLAB script in an M-file to simulate chips from suppliers S1 and S2. Do not worry about how the script works. Run it, and based on your output, tell which supplier you think has more reliable chips.
% Chips from suppliers S1 and S2.
%
NOS1 = 983; % Number of chips from S1 NOS2 = 871; % Number of chips from S2
NOWS1 = sum(rand(1,NOS1) >= 0.2); NODS1 = NOS1-NOWS1;
NOWS2 = sum(rand(1,NOS2) >= 0.3); NODS2 = NOS2-NOWS2;
Nmat = [ NOWS1 NOWS2; NODS1 NODS2 ] NOS = [ NOS1 NOS2 ]
fprintf(’Rel freq working chips from S1 is %4.2f.\n’,...
NOWS1/NOS1)
fprintf(’Rel freq working chips from S2 is %4.2f.\n’,...
NOWS2/NOS2)
52. If
N(Od,S1)
N(OS1) , N(OS1), N(Od,S2)
N(OS2) , and N(OS2) are given, compute N(Ow,S1) and N(Ow,S2) in terms of them.
53. If P(C) and P(B ∩C) are positive, derive the chain rule of conditional probability, P(A ∩ B|C) = P(A|B ∩C)P(B|C).
Also show that
P(A ∩ B ∩C) = P(A|B ∩C)P(B|C)P(C).
54. The university buys workstations from two different suppliers, Mini Micros (MM) and Highest Technology (HT). On delivery, 10% of MM’s workstations are defec-tive, while 20% of HT’s workstations are defective. The university buys 140 MM workstations and 60 HT workstations for its computer lab. Suppose you walk into the computer lab and randomly sit down at a workstation.
(a) What is the probability that your workstation is from MM? From HT?
(b) What is the probability that your workstation is defective? Answer: 0.13.
(c) Given that your workstation is defective, what is the probability that it came from Mini Micros? Answer: 7/13.
55. The probability that a cell in a wireless system is overloaded is 1/3. Given that it is overloaded, the probability of a blocked call is 0.3. Given that it is not overloaded, the probability of a blocked call is 0.1. Find the conditional probability that the system is overloaded given that your call is blocked. Answer: 0.6.
56. The binary channel shown in Figure 1.17 operates as follows. Given that a 0 is trans-mitted, the conditional probability that a 1 is received isε. Given that a 1 is transmit-ted, the conditional probability that a 0 is received isδ. Assume that the probability of transmitting a 0 is the same as the probability of transmitting a 1. Given that a 1 is received, find the conditional probability that a 1 was transmitted. Hint: Use the notation
Ti := {i is transmitted}, i = 0,1, and
Rj := { j is received}, j = 0,1.
Remark. Ifδ =ε, this channel is called the binary symmetric channel.
0 1
1− δ δ
0 1− ε 1
ε
Figure 1.17. Binary channel with crossover probabilitiesεandδ. Ifδ=ε, this is called a binary symmetric channel.
57. Professor Random has taught probability for many years. She has found that 80% of students who do the homework pass the exam, while 10% of students who don’t do the homework pass the exam. If 60% of the students do the homework, what percent of students pass the exam? Of students who pass the exam, what percent did the homework? Answer: 12/13.
58. A certain jet aircraft’s autopilot has conditional probability 1/3 of failure given that it employs a faulty microprocessor chip. The autopilot has conditional probability 1/10 of failure given that it employs a nonfaulty chip. According to the chip manufacturer, the probability of a customer’s receiving a faulty chip is 1/4. Given that an autopilot failure has occurred, find the conditional probability that a faulty chip was used. Use the following notation:
AF = {autopilot fails}
CF = {chip is faulty}.
Answer: 10/19.
59. Sue, Minnie, and Robin are medical assistants at a local clinic. Sue sees 20% of the patients, while Minnie and Robin each see 40% of the patients. Suppose that 60% of Sue’s patients receive flu shots, while 30% of Minnie’s patients receive flu shots and 10% of Robin’s patients receive flu shots. Given that a patient receives a flu shot, find the conditional probability that Sue gave the shot. Answer: 3/7.
60. You have five computer chips, two of which are known to be defective.
(a) You test one of the chips; what is the probability that it is defective?
(b) Your friend tests two chips at random and reports that one is defective and one is not. Given this information, you test one of the three remaining chips at random;
what is the conditional probability that the chip you test is defective?
(c) Consider the following modification of the preceding scenario. Your friend takes away two chips at random for testing; before your friend tells you the results, you test one of the three remaining chips at random; given this (lack of) informa-tion, what is the conditional probability that the chip you test is defective? Since you have not yet learned the results of your friend’s tests, intuition suggests that your conditional probability should be the same as your answer to part (a). Is your intuition correct?
1.6: Independence
61. (a) If two sets A and B are disjoint, what equation must they satisfy?
(b) If two events A and B are independent, what equation must they satisfy?
(c) Suppose two events A and B are disjoint. Give conditions under which they are also independent. Give conditions under which they are not independent.
62. A certain binary communication system has a bit-error rate of 0.1; i.e., in transmitting a single bit, the probability of receiving the bit in error is 0.1. To transmit messages,
a three-bit repetition code is used. In other words, to send the message 1, 111 is transmitted, and to send the message 0, 000 is transmitted. At the receiver, if two or more 1s are received, the decoder decides that message 1 was sent; otherwise, i.e., if two or more zeros are received, it decides that message 0 was sent. Assuming bit errors occur independently, find the probability that the decoder puts out the wrong message. Answer: 0.028.
63. You and your neighbor attempt to use your cordless phones at the same time. Your phones independently select one of ten channels at random to connect to the base unit.
What is the probability that both phones pick the same channel?
64. A new car is equipped with dual airbags. Suppose that they fail independently with probability p. What is the probability that at least one airbag functions properly?
65. A dart is repeatedly thrown at random toward a circular dartboard of radius 10 cm.
Assume the thrower never misses the board. Let Andenote the event that the dart lands within 2 cm of the center on the nth throw. Suppose that the Anare mutually independent and thatP(An) = p for some 0 < p < 1. Find the probability that the dart never lands within 2 cm of the center.
66. Each time you play the lottery, your probability of winning is p. You play the lottery n times, and plays are independent. How large should n be to make the probability of winning at least once more than 1/2? Answer: For p = 1/106, n≥ 693147.
67. Anne and Betty go fishing. Find the conditional probability that Anne catches no fish given that at least one of them catches no fish. Assume they catch fish independently and that each has probability 0< p < 1 of catching no fish.
68. Suppose that A and B are independent events, and suppose that A and C are indepen-dent events. If C⊂ B, determine whether or not A and B \C are independent.
69. Consider the sample space Ω = [0,1) equipped with the probability measure P(A) :=
A1 dω, A ⊂ Ω.
For A= [0,1/2), B = [0,1/4) ∪ [1/2,3/4), and C = [0,1/8) ∪ [1/4,3/8) ∪ [1/2,5/8)
∪[3/4,7/8), determine whether or not A, B, and C are mutually independent.
70. Given events A, B, and C, show that
P(A ∩C|B) = P(A|B)P(C|B) if and only if
P(A|B ∩C) = P(A|B).
In this case, A and C are conditionally independent given B.
71. Second Borel–Cantelli lemma.Show that if Bnis a sequence of independent events for which
∑
∞n=1P(Bn) = ∞,
then
P
∞
n=1
∞ k=n
Bk
= 1.
Hint: The inequality 1− P(Bk) ≤ exp[−P(Bk)] may be helpful.k 1.7: Combinatorics and probability
72. An electronics store carries three brands of computers, five brands of flat screens, and seven brands of printers. How many different systems (computer, flat screen, and printer) can the store sell?
73. If we use binary digits, how many n-bit numbers are there?
74. A certain Internet message consists of four header packets followed by 96 data pack-ets. Unfortunately, a faulty router randomly re-orders all of the packpack-ets. What is the probability that the first header-type packet to be received is the 10th packet to arrive?
Answer: 0.02996.
75. Joe has five cats and wants to have pictures taken of him holding one cat in each arm. How many pictures are needed so that every pair of cats appears in one picture?
Answer: 10.
76. In a pick-4 lottery game, a player selects four digits, each one from 0,...,9. If the four digits selected by the player match the random four digits of the lottery drawing in any order, the player wins. If the player has selected four distinct digits, what is the probability of winning? Answer: 0.0024.
77. How many 8-bit words are there with three ones (and five zeros)? Answer: 56.
78. A faulty computer memory location reads out random 8-bit bytes. What is the proba-bility that a random word has four ones and four zeros? Answer: 0.2734.
79. Suppose 41 people enter a contest in which three winners are chosen at random. The first contestant chosen wins $500, the second contestant chosen wins $400, and the third contestant chosen wins $250. How many different outcomes are possible? If all three winners receive $250, how many different outcomes are possible? Answers:
63 960 and 10 660.
80. From a well-shuffled deck of 52 playing cards you are dealt 14 cards. What is the probability that two cards are spades, three are hearts, four are diamonds, and five are clubs? Answer: 0.0116.
81. From a well-shuffled deck of 52 playing cards you are dealt five cards. What is the probability that all five cards are of the same suit? Answer: 0.00198.
82. A finite set D of n elements is to be partitioned into m disjoint subsets, D1,...,Dmin which|Di| = ki. How many different partitions are possible?
kThe inequality 1−x ≤ e−xfor x≥ 0 can be derived by showing that the function f (x) := e−x−(1−x) satisfies f(0) ≥ 0 and is nondecreasing for x ≥ 0, e.g., its derivative, denoted by f, satisfies f(x) ≥ 0 for x ≥ 0.
83. m-ary pick-n lottery. In this game, a player chooses n m-ary digits. In the lottery drawing, n m-ary digits are chosen at random. If the n digits selected by the player match the random n digits of the lottery drawing in any order, the player wins. If the player has selected n digits with k0zeros, k1ones, . . . , and km−1copies of digit m−1, where k0+ ··· + km−1= n, what is the probability of winning? In the case of n = 4, m= 10, and a player’s choice of the form xxyz, what is the probability of winning;
for xxyy; for xxxy? Answers: 0.0012, 0.0006, 0.0004.
84. In Example 1.46, what 7-bit sequence corresponds to two apples and three carrots?
What sequence corresponds to two apples and three bananas? What sequence corre-sponds to five apples?