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# Solution Manual For Probability And Stochastic Processes A Friendly Introduction For Electrical And Computer Engineers 3rd Edition By Roy D. Yates

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SAMPLE QUESTIONS

Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers (3rd Edition)

Chapter 1 Experiments, Models,

and Probabilities

Problems

Difficulty: Easy

1.1.1 Continuing Quiz 1.1, write Gerlandas

ent ire menu in words (supply prices

if you ivish).

1.1.2 For Gerlandas pizza in Quiz 1.1, answer

t hese questions:

(a) Are N and M mutually exclusive?

(b) Are N, T, and M collectively exhaust

ive?

( c) Are T and 0 mutually exclusive? State

t his condition in ivords.

( d) Does Gerlandas make Tuscan pizzas

with mushrooms and onions?

(e) Does Gerlandas make Neapolitan pizzas

t hat have neit her mushrooms nor

onions?

Moderate Difficu lt t Experts Only

1.1.3 Ricardos offers customers two kinds

of pizza crust, Roman (R) and Neapolitan

(N). All pizzas have cheese but not all pizzas

have tomato sauce. Roman pizzas can

have tomato sauce or t hey can be white

(W); Neapolitan pizzas always have tomato

sauce. It is possible to order a Roman pizza

with mushrooms (JV!) added. A Neapolitan

pizza can contain mushrooms or onions ( 0)

or both , in addition to t he tomato sauce and

cheese. Draw a venn diagram t hat shows

the relationship among t he ingredients N,

111, 0 , T, and W in t he menu of Ricardos

. . p1zzer1a.

1.2.1 A hypothet ical wi-fi transmission

can take place at any of three speeds

[

30 CHAPTER 1 EXPERIMENTS, MODELS, AND PROBABILITIES

d ep ending on t he condit ion of t he r adio

channel between a lap top and an access

point . The speeds are high (h) at 54 Mb/s,

med ium (m) at 11 ~1b/s, a nd low (l) at

1 lVIb/s. A user of t he wi-fi connection can

transmit a short signal corresponding to a

mouse click ( c), or a long sign al corresponding

to a tweet ( t) . Consider t he experiment

of monitoring \Vi-fi s ignals a nd obser ving

t he t r ansmission speed and t he length. An

observation is a t o-letter word, for example,

a high-speed, mouse-click transmission

is hm,.

(a) What is t he sample space of t he experi.

men t,?.

(b) Let _4.1 be t he event medium speed

connection. What are t he outcomes

1. i. n A 1 ?.

( c) Let A2 be t he event mouse click .

\i hat are t he outcomes in A2?

( d) Let A 3 be t he event hi gh speed

connection or low speed connection.

\i hat are t he outcomes in A 3?

( e) Are A1 , A2, and _43 mut ually exclusive?

( f) Are Ai, A2, and A3 collectively exhaust

ive?

1.2.2 _An integrated circu it factory has

three machines X, Y, a nd Z . Test one integrated

circuit produced by each machine.

Eit her a circuit is acceptab[e (a) or it fa ils

(f) . _An observation is a sequence of t hree

test results corresponding to t he circuits

from machines X, Y, and Z , respectively.

For example, aaf is t he observation t hat

t he circuits from X and Y pass t he test and

t he circuit from Z fails t he test.

(a) What are t he elements of t he sample

space of t his experiment?

(b) What are t he elements of t he sets

Z F = {circuit from Z fails} ,

XA = {circuit from X .is acceptable} .

(c) Are Zp and XA mut ua lly exclusive?

(d) Are Zp and XA collectively exhaust

ive?

( e) What a re t he elements of t he sets

C = {more t han one circuit acceptable},

D = {at least two circuits fa il}.

( f) Are C and D mutually exclusive?

(g) Are C and D collectively exhaustive?

1.2.3 Shuffle a deck of cards and turn over

t he first card. \i hat is t he sample space of

t his exper iment? How many outcomes are

in t he event t hat t he first card is a heart?

1.2.4 Find out t he birt hday (month and

day but not year) of a randomly chosen person.

\ i hat is t he sample space of t he exper

iment? Ho many outcomes are in t he

event t hat t he person is born in J uly?

1.2.5 T he sample space of an exper iment

consists of all undergr aduates at a university.

Give four examples of par t it ions.

1.2.6 T he sample space of an exper iment

consists of t he ineasured resistances of two

resistors . Give four exa1nples of part it ions.

1.3.1 Find P [BJ in each case:

(a) Events A and B ar e a pa rt ition and

l=> [AJ = 3P [BJ.

(b) For events A and B , P [A U BJ = P [AJ

and P [A n BJ = 0.

(c) For events _4. and B , P [A U BJ = P [AJl=>

[BJ.

1.3.2 You roll two fair six-sided dice; one

die is red , t he other is hite. Let Ri be t he

event that t he red die rolls i. Let vVj be t he

event t hat t he white die rolls j .

(a) What is P [R3 W2J?

(b) V hat is t he P [S5J t hat t he sum of t he

t o rolls is 5?

1.3.3 You roll two fair six-sided d ice.

Find t he probability P [D3J t hat t he absolute

value of t he difference of t he dice is 3.

1.3.4 Indicate hether each statement is

t rue or false.

(a) If P [-4.J = 21=>[Ac), t hen P [AJ = 1 / 2.

(b) For all A and B , P [ABJ < P [ AJ P [BJ.

[

( c) If P[A] < P[ BJ, then P [AB] < P[ BJ.

(d) If P[A n BJ = P[A], t hen P[A] > P[B].

1.3.5 Computer programs are classified by

the length of the source code and by the

execution t ime. l=>rograms with more than

150 lines in t he source code are b ig ( B ).

Programs \Vith < 150 lines are little (L).

Fast programs (F) run in less than 0.1 seconds.

Slow programs (W) r equire at least

0.1 seconds. l\/Ionitor a program executed

by a computer. Observe the length of the

source code and the run time. The probability

model for this experiment contains

the follo ing informat ion: P [LF] = 0.5,

P[BF] = 0.2, and P [BW] = 0.2. \i\!hat is

the sample space of the experiment? Calculate

the fo llo,ving probabilities: P [W], P[B],

and P[vV u BJ.

1.3.6 There are two types of cellu lar

phones, handheld phones (H) that you

carry and mobile phones (M) that are

mounted in vehicles. Phone calls cru1 be

classified by the traveling speed of the user

as fast (F) or slo (W). l\/Ionitor a cellular

phone call and observe the type of telephone

and the speed of the user. The probability

model for this experiment has the following

information: P [F] = 0.5, P [HF] = 0.2,

P[MW] = 0.1. Vhat is the sample space of

the experiment? Find the follo,ving probabilities

P [W], P [MF], and P [HJ.

1.3.7 Shuffle a deck of cards and turn over

the first card. What is the probability that

the first card is a heart?

1.3.8 You have a six-sided die that you

roll once and observe the number of dots

facing up,vards. hat is the sample space?

\i hat is the probability of each sample outcome?

\i hat is the probability of E, the

event that the roll is even?

1.3.9 A students score on a 10-point quiz

is equally likely to be any i11teger bet een

0 and 10. Vhat is the probability of an _4 ,

vhich requires the student to get a score

of 9 or more? \i hat is the probability the

student gets an F by getting less than 4?

PROBLEMS 31

1.3.10 Use Theorem 1.4 to prove the follo,

ving facts:

(a) P[A U BJ > P[-4]

(b) P [_4 uB] >P[B]

(c) P[A n BJ < P[A]

(d) P [A n BJ < P[B]

1.3.11 Use Theore1n 1.4 to prove by induction

the 7J,nion bound: For any collection

of events A1, , _4n,

n

P [A1 U A2 U U -4n] < LP [Ai] .

i=l

1.3.12 Using only t he three axioms of

probability, prove P = 0.

1.3.13 Using the three axioms of probability

and the fact that P [ 0 ] = 0, prove

Theorem 1.3. Hint: Define _4i = Bi for

i = 1, . . . , m and _4 i = 0 for i > 1n.

1.3.14 For each fact stated in Theorem

1.4, determine hich of the three axioms

of probability are needed to prove the

fact.

1.4.1 Mobile telephones perform handoffs

as they move from cell to cell. During a

call , a telephone either performs zero handoffs

(Ho), one handoff (H1), or more than

one handoff (H2 ) . In addition, each call is

either long ( L), if it lasts more than three

minutes, or brief ( B). The following table

describes the probabilities of the possible

types of calls .

L O.l 0.1 0.2

B 0.4 0.1 0.1

(a) What is the probability that a brief call

ill have no handoffs?

(b) \i hat is the probability that a call with

one handoff \Vill be long?

( c) hat is the probability that a long call

ill have one or more handoffs?

1.4.2 You have a six-sided die that you

roll once. Let Ri denote the event that

the roll is i. Let Gj denote the event that

[

32 CHAPTER 1 EXPERIMENTS, MODELS, AND PROBABILITIES

the roll is greater t han j. Let E denote

the event that the roll of the die is evennumbered.

(a) What is P[Rs lG1], the condit ional

probability t hat 3 is rolled given t hat

the roll is greater than 1?

(b) What is the conditional probability

that 6 is rolled given t hat the roll is

greater than 3?

( c) \i hat is P [Gs IE], t he conditional proba

b ili ty that the roll is greater than 3

given that the roll is even?

(d) Given that the roll is greater than 3,

what is the conditional probability that

the roll is even?

1.4.3 You have a shuffled deck of three

cards: 2, 3, and 4. You dra:v one card. Let

Ci denote the event t hat card i is picked.

Let E denote the event that the card chosen

is a even-numbered card.

(a) What is P[C2IE], the probability that

the 2 is picked given that an evennumbered

card is chosen?

(b) What is the conditional probability

that an even-numbered card is picked

given that the 2 is picked?

1.4.4 Phonesmart is having a sale on Bananas.

If you buy one Bana11a at full price,

you get a second at half price. Vhen couples

come in to buy a pair of phones, sales

of Apricots and Bananas are equally likely.

Moreover, given that the first phone sold

is a Banana, the second phone is twice as

likely to be a Banana rather than an Apricot.

What is the probability that a couple

buys a pair of Bananas?

1.4.5 The basic rules of genetics \Vere discovered

in mid-1800s by ~1endel , who found

that each characteristic of a pea plant, such

as hether the seeds \Vere green or yello,v,

is determined by two genes, one from each

parent. In his pea plants, Mendel found

that yello seeds \Vere a do1ninant trait over

green seeds. A yy pea with two yellow genes

has yellov seeds; a gg pea \Vith two recessive

genes has green seeds; a hybrid gy or yg

pea has yellov seeds. In one of Mendels experiments,

he started \Vith a parental generation

in which half the pea plants \Vere yy

and half the plants \Vere gg. The two groups

were crossbred so that each pea plant in the

first generation \Vas gy. In the second generation,

each pea plant \Vas equally likely

to inherit a y or a g gene from each firstgeneration

parent. V hat is the probability

P [Y] that a randomly chosen pea plant in

the second generation has yellov seeds?

1.4.6 Fi-om Problem 1.4.5, what is the

conditional probability of yy, that a pea

plant has two dominant genes given the

event Y that it has yellow seeds?

1.4.7 You have a shuffled deck of three

cards: 2, 3, and 4, and you deal out the

three cards. Let Ei denote the event that

ith card dealt is even numbered.

(a) \iVhat is P[E2 IE1], the probability t he

second card is even given that the first

card is even?

(b) Vhat is the conditional probability

that the first t o cards are even given

that the third card is even?

( c) Let Oi represent t he event that the ith

card dealt is odd numbered. What is

P[E2 I01], the conditional probability

that the second card is even given that

the first card is odd?

( d) \iVhat is the conditional probability

that the second card is odd given that

the first card is odd?

1.4.8 Deer t icks can carry both Lyme disease

and human granulocytic ehrlichiosis

(HGE). In a study of t icks in the ~1id,vest,

it was found t hat 16% carried Lyme disease,

10% had HGE, and that 10% of the

ticks that had either Ly1ne disease or HGE

carried both diseases.

(a) What is t he probability P [LH] that a

t ick carries both Lyme disease ( L) and

HGE (H)?

(b) \iVhat is the conditional probability

t hat a tick has HGE given that it has

Lyme disease?

[

1.5.1 Given the model of handoffs and call

lengt hs in Problem 1.4.1,

(a) What is the probability P[Ho) that a

phone makes no handoffs?

(b) What is t he probability a call is brief?

( c) \i hat is the probability a call is long or

there are at least two handoffs?

1.5.2 For the telephone usage model of

Example 1.18, let Brn denote the event that

a call is billed for m, minutes. To generate a

phone bill, observe t he duration of the call

in integer minutes (rounding up). Charge

for M minutes JV! = 1, 2, 3, .. . if the exact

duration T is M 1 < t < M. A more

complete probability model sho,vs that for

m, = 1, 2, . . . the probability of each event

Brri is

vhere a = 1 (0.57)113 = 0.171.

(a) Classify a call as long, L, if the call

lasts inore than three minutes. \i hat

is P [L)?

(b) What is the probabilitJr that a call will

be billed for nine minutes or less?

1.5.3 Suppose a cellular telephone is

equally likely to make zero handoffs (Ho),

one handoff (H1), or more t han one handoff

(H2). Also, a caller is either on foot ( F)

vith probability 5/12 or in a vehicle (V).

(a) Given t he preceding in.formation, find

three ways to fill in the fo llo,ving probability

table:

F

v

(b) Suppose ve also learn that 1/4 of all

callers are on foot inaking calls with no

handoffs and that 1 /6 of all callers are

vehicle users making calls vi th a single

handoff. Given these additional facts ,

find all possible ways to fill in the table

of probabilities.

PROBLEMS 33

1.6.1 Is it possible for A and B to be independent

events yet satisfy A = B?

1.6.2 Events A and B are equiprobable,

mutually exclusive, and independent.

What is P[A]?

1.6.3 At a P honesmart store, each phone

sold is twice as likely to be an Apricot as a

Banana. _Also each phone sale is independent

of any other phone sale. If you monitor

the sale of tvo phones, what is the probability

that the two phones sold are the same?

1.6.4 Use a \ ! enn diagram in vhich the

event areas are proportional to t heir probabilities

to illustrate tvo events A and B

that are independent.

1.6.5 In an experiment, A and B are mutually

exclusive events vith probabilities

P [A) = 1/4 and P[B) = 1/8.

(a) Find P[A n BJ, P [A u BJ, P [A n Be],

and P[A U Be) .

(b) Are A and B independent?

1.6.6 In an experiment, C and D are independent

events with probabilities P [C) =

5/8 and P[D) = 3/8 .

(a) Determine the probabilities P[C n DJ,

f>[C n D e), and P [Cc n De).

(b) Are cc and De independent?

1.6.7 In an experiment, A and B are mutually

exclusive events vith probabilities

P [A U BJ = 5/8 and P[A) = 3/8.

(a) F ind l=> [B], P[A n Be], and P[A U Be).

(b) Are A and B independent?

1.6.8 In an experiment, C, and D

are independent events with probabilities

P [C n DJ = 1/3, and P [C) = 1/2.

(a) F ind P[D], P[C n De], and P [Cc n D e).

(b) F ind P [C uD) and P [C u Dc).

( c) _A.re C and De independent?

1.6.9 In an experiment with equiprobable

outcomes, the sample space is S =

{1, 2, 3,4} andP[s] = l/4forall s ES.

Find three events in S that are pair,vise independent

but are not independent. (Note:

[

www.ebook3000.com

34 CHAPTER 1 EXPERIMENTS, MODELS, AND PROBABILITIES

Pair,vise independent events meet the first

three conditions of Definition 1. 7).

1.6.10 (Continuation of Problem 1.4.5)

One of rvlendel s most s ignificant results

vas the conclusion that genes determining

different characteristics are transmitted

independently. In pea plants, l\/Iendel

found that round peas (r) are a dominant

trait over vrinkled peas ( UJ). Mendel

crossbred a group of (rr, yy) peas with a

group of ( ?lJUJ ,gg) peas. In t his notation,

rr denotes a pea with two ((round genes

and ?1J?1J denotes a pea with tvo wrinkled

genes. The first generation vere either

(r1D,yg) , (r1D ,gy) , (1Dr, yg), or (v1r, gy)

plants vith both hybrid shape and hybrid

color. Breeding among the first generation

yielded second-generation plants in

vhich genes for each characteristic were

equally likely to be either dominant or recessive.

Vhat is the probability P [Y] that

a second-generation pea plant has yellov

seeds? What is the probability P [R] that

a second-generation plant has round peas?

Are R and Y independent events? How

many visib ly different kinds of pea plants

would l\/Iendel observe in the second generation?

Vhat are the probabilities of each

of these kinds?

1.6.11 For independent events A and B ,

prove that

(a) A and B e are independent.

(b) Ac and B are independent.

( c) Ac and B c are independent.

1.6.12 Use a Venn d iagram in which the

event areas are proportional to their probabilities

to illustrate three events A, B , and

C that are independent.

1.6.13 use a Venn diagram in which event

areas are in proportion to their probabilities

to illustrate events _4, B, and C that are

pair,vise independent but not independent.

1.7.1 Follo,ving Quiz 1.3, use 1VIATLAB,

but not the r andi function, to generate

a vector T of 200 independent test scores

such that all scores bet,veen 51and100 are

equally likely.

Chapter  2 Sequential Experiments

Problems

Difficulty: Easy

2.1 .1 Suppose you flip a coin tvice. On

any flip , the coin comes up heads with probability

1/4. Use Hi and Ti to denote the

result of flip i.

(a) What is t he probability, P[H1 IH2], that

the first flip is heads given that the second

flip is heads?

(b) What is the probabilit:y that t he first

flip is heads and the second flip is tails?

2.1 .2 For Example 2.2, suppose P[G1) =

1/ 2, P[G2 IG1) = 3/4, and P[G2 IR1] = 1/4.

Find P[G2), P[G2 IG1), and P[G1 IG2).

2.1 .3 At the end of regulation time, a basketball

team is trailing by one point and a

player goes to the line for t o free throvvs.

If the player inakes exactly one free throw,

the game goes into overtime. The probability

that the first free throw is good is

1/ 2. However , if the first attempt is good,

the player relaxes and the second attempt is

good \Vi th probability 3 / 4. However, if the

player misses the first attempt, the added

pressure reduces the success probability to

1/4. What is the probability that the game

goes into overtime?

2.1 .4 You have t o biased coins. Coin A

comes up heads with probability 1/4. Coin

B comes up heads ith probability 3/4.

However, you are not sure which is \Vhich,

so you choose a coin randomly and you flip

1. If t he flip is heads, you guess that the

flipped coin is B; otherwise, you guess that

t he flipped coin is .4. Vhat is the probability

P[C) that your guess is correct?

Moderate Difficu lt Experts Only

2.1.5 Suppose that for the general populat

ion, 1 in 5000 people carries the human immunodeficiency

virus (HIV). A test for the

presence of HIV yields either a positive ( +)

or negative (-) response. Suppose t he test

gives the correct ans,ver 993 of the t ime.

What is P[- IHJ , the conditional probability

that a person tests negative given that

the person does have the HIV virus? What

is P[HI+], the condit ional probability that

a randomly chosen person has the HIV virus

given that the person tests positive?

2.1.6 A machine produces photo detectors

in pairs. Tests show that the first photo

detector is acceptable with probability 3 /5.

When the first photo detector is acceptable,

the second photo detector is acceptable

with probability 4/5. If the first photo

detector is defective, the second photo detector

is acceptable ith probability 2/5.

(a) F ind the probability that exactly one

photo detector of a pair is acceptable.

(b) Find the probability t hat both photo

detectors in a pair are defective.

2.1.7 You have two biased coins. Coin .4

comes up heads \Vith probability 1/ 4. Coin

B comes up heads with probability 3/4.

Ho,vever , you are not sure which is which

so you flip each coin once, choosing the first

coin randomly. Use Hi and Ti to denote the

result of flip i. Let .41 be the event t hat coin

A was flipped first. Let B1 be the event that

coin B was flipped first. Vhat is P[H1H2)?

[

58 CHAPTER 2 SEQUENTIAL EXPERIMENTS

Are H 1 and H 2 independent? Explain your

answer.

2 .1. 8 A particular birth defect of the heart

is rare; a ne,vborn infant will have t he defect

D vith probability P[D) = 10- 4

. In

the general exa1n of a ne,vborn, a particular

heart arrhythmia A occurs with probability

1. 99 in infants vi th the defect. However,

the arrhythmia also appears ,;vith probability

0.1 in infants without the defect. \!\!hen

the arrhythmia is present, a lab test for the

defect is performed. The result of the lab

test is either positive (event r+) or negative

(event T ). In a newborn vith the defect,

the lab test is positive vith probability

p = 0.999 independent from test to test.

In a ne,vborn ,;vithout the defect , the lab

test is negative vith probability p = 0.999.

If the arrhythmia is present and the test

is positive, then heart surgery (event H) is

performed.

(a) Given the arryth1nia A is present, vhat

is the probability the infant has the defect

D?

(b) Given that an infant has the defect,

what is the probability P [H IDJ that

heart surgery is performed?

( c) Given that the infant does not have

the defect, what is t he probability

q = P [HIDc) t hat an unnecessary heart

surgery is performed?

(d) F ind the probability P[H) that an inf

ant has heart surge1y performed for

the arrythmia.

( e) Given that heart surgery is performed,

what is the probability that the newborn

does not have the defect?

2.1 .9 Suppose Dagwood (Blondies husband)

wants to eat a sandwich but needs to

go on a diet. Dagwood decides to let the flip

of a coin determine vhether he eats. using

an unbiased coin, Da~vood will postpone

the diet (and go directly to the refrigerator)

if eit her (a) he flips heads on his first flip or

(b) he flips tails on the first flip but then

proceeds to get tvo heads out of the next

three flips. Note that the first flip is not

counted in the attempt to win tvo of three

and that Dag,vood never performs any unnecessary

flips. Let Hi be the event that

Dag,vood flips heads on try i. Let Ti be the

event t hat tails occurs on flip i.

(a) Draw the tree for this experiment. Label

t he probabilities of all outcomes.

(b) \i hat are P [H3) and P [T.1)?

( c) Let D be the event t hat Dag,vood must

diet. What is P[D)? \i\!hat is P[H1IDJ?

( d) Are H 3 and H 2 independent events?

2.1.10 The quality of each pair of photo

detectors produced by the machine in Problem

2.1.6 is independent of the quality of

every other pair of detectors.

(a) \!\!hat is the probability of finding no

good detectors in a collection of n pairs

produced by the machine?

(b) How many pairs of detectors must the

machine produce to reach a probability

of 0.99 that there vill be at least one

acceptable photo detector?

2.1.11 In Steven Strogatzs New York

Times blog http: I I opinionator. blogs.

nytirnes.corn/2010/04/25/chances-are/

?ref=opinion, the follo,ving problem vas

posed to highlight the confusing character

of conditional probabilities.

Before going on 1;acation for a 71Jeek, you

ask yo1J,r spacey friend to 71Jater yo1J,r ailing

plant. Without 111ater, the plant has a 90

percent chance of dying. E1;en 71Jith proper

111atering, it has a 20 percent chance of dying.

And the probability that your friend

1Dill forget to 71Jater it is 30 percent. (a)

Whats the chance that yo1J,r plant 7Dill survive

the 111eek? {b) If its dead 71Jhen you

return, 71Jhat s the chance that your friend

forgot to 71Jater it? ( c) If yo1J,r friend forgot

to 11Jater it, 71Jhat s the chance itll be dead

1Dhen you return?

Solve parts (a), (b) and (c) of t his problem.

2.1.12 Each t ime a fishe1man casts his

line, a fish is caught ,;vith probability p, independent

of vhether a fish is caught on

any other cast of t he line. The fisherman

will fish a ll day until a fish is caught and

[

then he vill quit and go home. Let Ci denote

the event that on cast i the fisherman

catches a fish. Draw the tree for this experiment

and find P[C1), P[C2], and P[Cn] as

funct ions of p.

2.2.1 On each turn of the knob, a gumball

machine is equally likely to dispense a

red, yellow, green or blue gumball, independent

from turn to turn. After eight turns,

what is the probability I>[R2Y2G2B2] that

you have received 2 red, 2 yellow, 2 green

and 2 blue gumballs?

2.2.2 A Starburst candy package contains

12 individual candy pieces. Each piece is

equally likely to be berry, orange, lemon, or

cherry, independent of all other pieces.

(a) What is the probability that a Starburst

package has only berry or cherry

pieces and zero orange or lemon pieces?

(b) What is the probability that a Starburst

package has no cherry pieces?

( c) \i hat is t he probability P [F1] that all

twelve pieces of your Star burst are the

same flavor?

2.2.3 Your Starburst candy has 12 pieces,

three pieces of each of four flavors: berry,

le1non, orange, and cherry, arranged in a

random order in the pack. You draw the

first three pieces from the pack.

(a) What is the probability they are all t he

same flavor?

(b) What is the probability they are all different

flavors?

2.2.4 Your Starburst candy has 12 pieces,

three pieces of each of four flavors: berry,

lemon, orange, and cherry, arranged in a

random order in the pack. You draw the

first four pieces from the pack.

(a) What is t he probability P[F1] they are

all t he same flavor?

(b) What is t he probability P [ F4] they are

all different flavors?

( c) \i hat is the probability P [ F2] that your

Star burst has exactly two pieces of each

of tvo different flavors?

PROBLEMS 59

2.2.5 In a game of rummy, you are dealt

a seven-card hand.

(a) What is the probability P[R7 ] that your

hand has only red cards?

(b) \i hat is the probability P [ F] that your

hand has only face cards?

(c) V hat is t he probability P[R1F] that

your hand has only red face cards?

(The face cards are jack, queen, and

king.)

2.2.6 In a game of poker, you are dealt a

five-card hand.

(a) hat is the probability I>[R5] that your

hand has only red cards?

(b) \i hat is the probability of a full

house with three-of-a-kind and two-ofa-

kind?

2.2. 7 Consider a binary code vi th 5 bits

(0 or 1) in each code vord. An example

of a code word is 01010. How many different

code words are there? Hov many code

words have exactly three Os?

2.2.8 Consider a language containing four

letters: A, B, C, D. Hov many three-letter

words can you form in this language? Hov

many four-letter vords can you form if each

letter appears only once in each word?

2.2.9 On an American League baseball

team vith 15 field players and 10 pitchers,

the manager selects a starting lineup with

8 field players, 1 pitcher, and 1 designated

hitter. The lineup specifies the players for

these positions and the positions in a batting

order for the 8 field players and designated

hitter. If t he designated hitter must

be chosen among all t he field players, how

many possible starting lineups are there?

2.2.10 Suppose that in Proble1n 2.2.9, the

designated hitter can be chosen from among

all the players. How many possible starting

lineups are there?

2.2.11 At a casino, the only game is numberless

roulette. On a spin of the vheel,

the ball lands in a space with color red ( r),

green (g), or black ( b). The wheel has 19 red

spaces, 19 green spaces and 2 black spaces.

[

60 CHAPTER 2 SEQUENTIAL EXPERIMENTS

(a) In 40 spins of the wheel, find t he probabili

ty of the event

A= {19 reds, 19 greens, and 2 blacks} .

(b) In 40 spins of the vheel , find the probability

of G19 = {19 greens}.

( c) The only bets a llowed are red and

green. Given that you randomly choose

to bet red or green, vhat is t he probability

p that your bet is a vvinner?

2.2.12 A basketball team has three pure

centers, four pure for ards, four p1ue

guards, and one swingman who can p lay

either guard or forward. A pure posit ion

p layer can play only the designated posit

ion. If the coach must start a lineup with

one center, t o for,vards, and two guards,

how inany possible lineups can the coach

choose?

2.2.13 An instant lottery t icket consists

of a collection of boxes covered with gray

\Vax. For a subset of the boxes, the gray wax

hides a special mark. If a p layer scratches

off the correct nu1nber of the marked boxes

(and no boxes vithout the mark) , then that

ticket is a \Vinner. Design an instant lottery

game in vhich a player scratches five boxes

and the probability that a ticket is a inner

is approximately 0.01.

2.3.1 Consider a binary code vith 5 bits

(0 or 1) in each code ord. An example of

a code word is 01010. In each code word a bit is a zero with probability 0 .8, independent

of any other bit.

(a) What is the probability of the code

word 00111?

(b) What is the probability t hat a code

word contains exactly three ones?

2.3.2 T he Boston Celtics have won 16

NBi\. championships over approximately 50

years. Thus it may seem reasonable to assume

that in a given year the Celt ics \Vin

the t it le \Vith probability p = 16/50 = 0.32,

independent of any other year. Given such

a model, what \Votlld be the probability

of the Celt ics winning eight straight championships

beginning in 1959? Also, what

would be the probability of the Celtics winning

the t it le in 10 out of 11 years, starting

in 1959? Given your answers, do you trust

this simple probability model?

2.3.3 Suppose each day that you drive to

work a traffic light that you encounter is either

green \Vith probability 7 /16, red with

probability 7 / 16, or yello vith probability

1/8, independent of the status of the liaht 0

on any other day. If over the course of five

days, G, Y, and R denote the number of

times the light is found to be green, yello,v,

or red, respectively, hat is the probability

that P [G = 2, Y = 1, R = 2]? _Also , vhat is

the probability P [G = R]?

2.3.4 In a game between t o equal teams,

the home team \Vins \Vith probability p >

1/ 2. In a best of t hree playoff series, a

team vith the home advantage has a game

at home, fo llowed by a game a ay, fo llowed

by a home game if necessary. The series is

over as soon as one team ins t o games.

Vhat is P [H], the probability t hat the team

with the ho1ne advantage wins the series? Is

the home advantage increased by playing a

three-game series rather than a one-game

playoff? That is, is it true that P [HJ > p

for all p > 1/2?

2.3.5 A collection of field goal kickers are

divided into groups 1 and 2. Group i has

3i kickers. On any kick, a kicker fro1n

group i vvill kick a fie ld goal with probability

1/(i +l), independent of the outcome

of any other kicks.

(a) A kicker is selected at random from

among all the kickers and attempts one

field goal. Let K be the event that a

fie ld goal is kicked. F ind P [ K].

(b)

(c)

Tvo kickers are selected at random J{

J

is the event that kicker j kicks a field

goal. Are J{ i and J{ 2 independent?

_A. kicker is selected at random and attempts

10 fie ld goals. Let M be the

number of inisses. F ind P [M = 5].

[

2.4.1 A particular operation has s ix components.

Each component has a fai lure

probability q, independent of a ny other

component. A successful operation requires

both of t he fo llowing condit ions:

• Components 1, 2, and 3 all \Vork, or

component 4 \Vorks.

• Component 5 or component 6 works.

Drav a block diagram for this operation

similar to those of F igure 2.2 on page 53.

Derive a formula for t he probability P[W]

t hat the operation is successful.

2.4.2 We wish to modify t he cellular telephone

coding system in Example 2.21 in

order to reduce the number of errors. In

particular, if there are t o or t hree zeroes

in t he received sequence of 5 bits, \Ve ill

say that a deletion (event D) occurs. Ot her,

vise, if at least 4 zeroes are received, t he

receiver decides a zero \Vas sent, or if at least

4 ones are received , the receiver decides a

one was sent. We say t hat an error occurs

if i \Vas sent and the receiver decides j f=. i

\Vas sent. For t his modified protocol, hat

is the probability P [E] of an error? What

is the probability P[D] of a deletion?

2.4.3 Suppose a 10-digit phone number is

transmitted by a cellular phone using four

binary symbols for each d igit, using t he

model of binary symbol errors and deletions

given in Problem 2.4.2. Let C denote the

number of bits sent correctay, D t he number

of deletions, and E the number of errors.

Find P[C = c, D = d, E = e] for all c,

d, and e.

2.4.4 Consider the dev ice in Problem

2.4. l. Suppose we can replace any one

component ith an ultrareliable component

that has a failure probability of q/2 = 0.05.

\i\Thich component should we replace?

2.5 .1 Build a IVIATLAB simulation of 50

trials of t he experiment of Example 2.3.

Your output should be a pair of 50 x 1 vectors

C and H . For the ith trial, Hi will

PROBLEMS 61

record vhether it \Vas heads (Hi = 1) or

tails (Hi = 0), and Ci E { 1, 2} \Vill record

which coin \Vas picked.

2.5.2 Following Quiz 2.3, suppose the

communication link has different error

probabilities for trans1nitting 0 and 1.

Vhen a 1 is sent, it is received as a 0 with

probability 0.01. V hen a 0 is sent, it is received

as a 1 vi th probability 0 .03. Each

bit in a packet is still equally likely to be a

0 or 1. Packets have been coded such t hat if

five or fewer bits are received in error, t hen

the packet can be decoded. Simulate the

transmission of 100 packets, each containing

100 bits. Count the number of packets

decoded correctly.

2.5.3 For a failure probability q = 0.2,

simulate 100 tria ls of the s ix-component

test of I>rob lem 2.4. l. Ho many devices

were found to work? Perform 10 repetitions

of the 100 trials. What do you learn from

10 repetitions of 100 trials com pared to a

simulated experiment vvith 100 trials?

1. 5 .4 \i rite a JVIA TLAB function

N=countequal(G,T)

that duplicates the action of h i st (G, T) in

Example 2.26. Hint : Use ndgri d.

2.5.5 In this problem, \Ve use a MATLAB

simulation to solve Problem 2.4.4. Recall

that a particular operation has six components.

Each component has a failure probability

q independent of any other component.

The operation is successful if both

• Components 1, 2, and 3 a ll work, or

component 4 orks.

• Component 5 or component 6 \Vorks.

V ith q = 0.2, simulate the replacement of

a component \Vith an ultrareliable component.

For each replacement of a regular

component, perform 100 trials. Are 100

trials sufficient to decide which component

should be replaced?

Chapter  3 Discrete Random Variables

Problems

Difficulty: Easy

3.2.1 The random variable fl has P l\!IF

P ( )

_ { c(l / 2)71 n, = 0, 1, 2,

N Tl,

0 otherwise.

(a) What is t he value of t he constant c?

(b) What isP[N< l )?

3.2.2 The random variable V has Pl\IIF

Pv(v) = { cv

2

v = 1, 2, 3,4,

0 other,vise.

(a) F ind t he value of t he constant c.

(b) F ind P [V E { ?.L

2 11.l= 1, 2, 3, .. }).

( c) Find t he probability t hat V is even.

(d) F ind P [I > 2).

3.2.3 The random variable X has Pl\!IF

P ( )

_ { c/ x ::i; = 2, 4, 8,

xx

0 ot herwise.

(a) What is t he value of t he constant c?

(b) What is P [X = 4)?

( c) \ i hat is P [X < 4)?

(d) What is P [3 < X < 9)7

3.2.4 In each at-bat in a baseba ll game,

mighty Casey s ings at every pitch. T he

Moderate Difficu lt Experts Only

result is eit her a home run (vith probability

q = 0.05) or a strike. Of course, t hree

strikes and Casey is out .

(a) What is t he p robability P [H ) t hat

Casey hits a ho1ne run?

(b) For one at-bat, vhat is t he Pl\IIF of fl,

t he number of t imes Casey s ings his

bat?

3.2.5 A tablet computer t r a nsmits a file

over a \Vi-fi link to an access point. Depending

on t he s ize of t he file, it is t r ansmitted

as N packets where N has PMF

P ( )

{ c/n, n= l , 2,3,

JV Tl,

0 otherwise.

(a) F ind t he constant c .

(b) Vhat is t he probability t hat N is odd?

( c) Each packet is received correctly with

probability p, and t he file is received

correctly if all N packets are received

correctly. Find P [CJ, t he probability

t hat t he file is received correctly.

3.2.6 In college basketball, vhen a player

is fouled while not in t he act of shooting

and t he opposing team is in t he pena lty,

t he player is awarded a l and l . In t he 1

and 1, t he player is awarded one free t hro ,

and if t hat free t hrow goes in t he player

is awarded a second free t hrow. F ind t he

PMF of Y , t he number of points scored in

[

a 1 and 1 given t hat any free throv1 goes

in vith probability p, independent of any

other free t hrow.

3.2. 7 You roll a 6-sided die repeatedly.

Starting with roll i = 1, let Ri denote the

result of roll i. If Ri > i, t hen you will roll

again; otherwise you stop. Let N denote

t he number of rolls.

(a) What is P [N > 3]?

(b) F ind the PlVIF of J\T.

3.2.8 v ou are manager of a t icket agency

t hat sells concert t ickets. You assume that

people vill call three times in an attempt

to buy t ickets and then give up. You vvant

to make sure that you are able to serve at

least 953 of t he people vho vant t ickets.

Let p be the probability that a caller gets

t hrough to your t icket agency. \i hat is the

minimum value of p necessary to meet your

goal?

3.2.9 In the t icket agency of Proble1n

3.2.8, each telephone ticket agent is

available to receive a call with probability

0.2. If a ll agents are busy when someone

calls, t he caller hears a busy signal. \i hat

is the minimum number of agents that you

have to hire to meet your goal of serving

953 of t he custo1ners vho vant t ickets?

3.2.10 Suppose when a baseball player

gets a hit, a single is twice as likely as a

double, vhich is twice as likely as a triple,

vhich is tvice as likely as a home run. Also,

t he players batting average, i.e., the probability

the player gets a hit, is 0.300. Let B

denote the number of bases touched safely

during an at-bat. For example, B = 0 vvhen

t he player makes an out, B = 1 on a single,

and so on. \i hat is t he f> lVIF of B?

3.2.11 \i hen someone presses SEND on

a cellular phone, t he phone attempts to set

up a call by transmitting a SET.UP message

to a nearby base station. The phone waits

for a response , and if none arrives wit hin

0.5 seconds it tries again. If it doesnt get a

response after n, = 6 tries, the phone stops

transmitting messages and generates a busy

signal.

PROBLEMS 107

(a) Draw a tree diagram t hat describes the

call setup procedure.

(b) If all transmissions are independent

and the probability is p that a SETUP

message will get through, vhat is the

PMF of K , the number of messages

trans1nitted in a call attempt?

( c) \i hat is the probability that the phone

will generate a busy signal?

( d) As manager of a cellular phone system,

you vant the probability of a busy signal

to be less than 0.02. If p = 0.9,

vhat is the minimum value of n necessary

to achieve your goal?

3.3.1 In a package of lVI&Ms, Y, the

number of yellow M&~1Is , is uniformly distributed

bet,veen 5 and 15.

(a) Vhat is t he P~!IF of Y?

(b) \i hatisP[Y

( c) \i hat is P [Y > 12]?

( d ) \iVhat is P [8 < Y < 12]?

3.3.2 In a bag of 25 ~1I&Ms, each piece

is equally likely to be red, green, orange,

blue, or bro,vn, independent of t he color of

any other piece. F ind the the PMF of R,

the number of red pieces. \i hat is the probability

a bag has no red M&~lfs?

3.3.3 \i hen a conventional paging system

transmits a message, the probability that

the message will be received by t he pager

it is sent to is p. To be confident that a

message is received at least once, a system

transmits t he message n, times.

(a) _Assuming all transmissions are independent,

vhat is the PMF of K, the

number of times t he pager receives the

same message?

(b) Assume p = 0.8. What is the minimum

value of n that produces a probability

of 0.95 of receiving the message at least

once?

3.3.4 You roll a pair of fair dice unt il

you roll doubles (i.e., both dice are the

same). \iVhat is t he expected number, E[N],

of rolls?

[

108 CHAPTER 3 DISCRETE RANDOM VARIABLES

3.3.5 \i hen you go fishing, you attach 1n

hooks to your line. Each t ime you cast your

line, each hook will be sv;,rallo,ved by a fish

vi th probability h, independent of whether

any other hook is svallowed. What is t he

PMF of I<, t he number of fish t hat are

hooked on a single cast of t he line?

3.3.6 Any time a child t hrows a Frisbee,

t he childs dog catches t he Frisbee with

probability p, independent of whether t he

Fr isbee is caught on any previous t hrow.

\ i hen t he dog catches t he F risbee, it runs

avay vith t he Frisbee, never to be seen

again. The child cont inues to t hrov t he

Fr is bee unt il t he dog catches it . Let X

d enote t he number of times t he F risbee is

thrown.

(a) What is t he PMF Px(x)?

(b) If p = 0.2, what is t he probability t hat

t he child vill t hrow t h e F r isbee more

t han four t imes?

3.3. 7 \ i h en a tvo-,vay p aging syste1n

transmits a message, t he probability t hat

t he message vill be received by t he pager it

is sent to is p. When t he pager receives t he

message, it t r ans1nits an acknowledgment

signal (ACK) to t he paging system. If t he

paging system does not receive t he ACK, it

sends t he message again.

(a) What is t he PMF of N, t he number of

times t he syste1n sends t he same message?

(b) The paging co1npany vants to li1nit t he

number of t imes it has to send the same

message. It has a goal of P [ N < 3] >

0.95. Vhat is t he minimum value of p

necessary to achieve t he goal?

3.3.8 The number of bytes B in an

HTML file is t he geometr ic (2.5 10- 5)

r andom variable. \i hat is t he probability

P[B > 500,000] t hat a file has over 500 ,000

bytes?

3.3.9

(a) Star ting on day 1, you b uy one lottery

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