1.1-4. Some ornithologists were interested in the clutch size of the common gallinule. They observed the number ofeggs in each of 117 nests, yielding the following data:

7 5 13 7 7 8 9 9 9 8 8 9 9 7 75 9 7 7 4 9 8 8 10 9 7 8 8 8 79 7 7 10 8 7 9 7 10 8 9 7 11 10 9948689998858899

14 10 8 9 9 9 8 7 9 7 9 10 10 7 611 7 7 6 9 7 7 6 8 9 4 6 9 8 97 9 9 9 9 8 8 8 9 9 9 8 10 9 9857876777659

(a) Construct a frequency table for these data.(b) Draw a histogram.

{;) (c) What is the mode (the typical clutch size)?L S. Noticing that some of the gallinules in the last exercise had a second brood during the summer, the ornithologists

became interested in comparing the clutch sizes for the second brood with those for the first brood. They wereable to collect the following clutch sizes for the second brood:

446559656979486598 8 7 6 8 8 9 10 9 9 7 4 6 8 7 5

(a) Construct a frequency table for these data.(b) Draw a histogram.(c) Is there a typical clutch size for second broods?

LI-6. Before buying enough cereal to obtain a set of four prizes (Example 1.1-4), a family decided to use simulationto estimate the number of boxes it would have to buy. Each member of the family rolled a four-sided die untilhe or she observed each face at least once. The family members repeated this exercise 100 times and recordedthe numbers of rolls needed as follows:

8 6 6 4 13 11 19 13 4 15 8 13 5 8 916 9 5 12 6 4 8 6 6 6 11 6 5 10 56 5 8 4 5 14 5 7 5 7 5 8 4 9 89 13 8 14 5 24 4 5 6 7 5 4 8 7 6

11 4 5 6 5 10 10 4 5 14 10 15 6 9 87 10 14 10 8 8 10 7 9 10 8 10 9 7 5

12 7 16 6 5 11 4 11 5 5

(a) Construct a frequency table for these data.() (b) Draw a histogram.

Ll~'.'j For a Texas lottery game, Cash Five, 5 integers are selected randomly out of the first 37 positive integers. Thefollowing table lists the numbers of odd integers out of each set of five integers selected in 100 consecutivedrawings in 2007:

3 2 2 2 2 3 3 2 4 2 4 2 2 3 2 3 1 3 2 34 3 3 4 5 3 2 1 2 0 2 2 2 3 1 2 1 3 3 12 5 4 0 3 3 0 1 3 3 3 5 2 2 3 1 4 4 3 34 3 3 1 2 3 1 3 2 3 3 2 2 3 3 4 3 2 1 53 0 1 2 2 4 3 3 3 1 2 4 4 1 1 4 4 1 1 3

(a) Construct a frequency table for these data.(b) Draw a histogram.

Ll~Consider a bowl containing 10 chips of the same (a) Assign a reasonable p.m.f. f(x) to the out-rJ size such that 2 are marked "one," 3 are marked come space.::two,':,3 are marke~ "three," and 2 are marked (b) Simulate this experiment at least n = 100four. Select a chip at random and read the times and find the relative frequency his-

number. Here S = {1.2,3,4}. togram h(x). HINT: Here you can use a com-

1.

L

1.

1.

puter to perform the simulation; or simplyuse the table of random numbers (Table IXin the appendix), start at a random spot,and let an integer in the set {O,1} = 1, in{2,3,4} = 2, in {5,6,7} = 3, in {8,9} =4.

(c) Plot/(x) and h(x) on the same graph.

.o/;J Toss two coins at random and count the numberr; of heads that appear "up." Here S = {O,I,2}.ln

Chapter 2, we discover that a reasonable prob-ability model is given by the p.m.f. /( 0) =1/4,/(1) = 1/2,/(2) = 1/4. Repeatthisexperi-ment at least n = 100 times, and plot the resultingrelative frequency histogram h(x) on the samegraph with/(x).

(c)

istsere 1.1.10. Let the random variable X be the number of

tosses of a coin needed to obtain the first head.Here S = {1,2,3,4,...}. In Chapter2, we findthat a reasonable probability model is given bythe p.m.f. f(x) = (1/2 )Z, xES. Do this experi-ment a large number of times, and compare theresulting relative frequency histogram h(x) withf(x).

1.1-lL In 1985, AI Bumbry of the Baltimore Orioles andDarren Brown of the Minnesota Twins had thefollowing numbers of hits (H) and official at bats(AB) on grass and artificial turf:

ionnilljed

Playing Surface Bumbry Brown

GrassArtificial TurfTotal

(a) Find the batting averages BA (namely,H/AB) of each player on grass.

(b) Find the BA of each player on artificial turf.(c) Find the season batting averages for the two

players.(d) Interpret your results.

Ll-12. In 1985, Kent Hrbek of the Minnesota Twins andDion James of the Milwaukee Brewers had thefollowing numbers of hits (H) and official at bats(AB) on grass and artificial turf:

Thelive

1.2 PROPERTIES OF PROBABILITY

Section 1.2 Properties of Probability 11

Playing Surface Hrbek JamesAB H BA AB H BA

Grass 204 SO 329 93Artificial Tud 355 124 S8 21Total 559 174 387 114

(a) Find the batting averages BA (namely,H/AB) of each player on grass.

(b) Find the BA of each player on artificial turf.(c) Find the season batting averages for the two

players.(d) Interpret your results.

.1-13. If we had a choice of two airlines, we would pos-sibly choose the airline with the better "on-timeperformance." So consider Alaska Airlines andAmerica West, using data reported by ArnoldBarnett (see references):

Alaska AmericaAirline Airlines West

Relative RelativeFrequency Frequency

Destination on Time on Time

497559

694811

Los Angeles

Phoenix

San Diego

San Francisco

Seattle

4840

5255383

448

221

233

212

232

503

6051841

2146

AB H BA AB H BA

295 77 92 1849 16 168 53 320

449201262

344 93 260 71

3274 6438- -

ms 7225Five-City Total

(a)

(b)

(c)

For each of the five cities listed, which airlinehas the better on-time performance?Combining the results, which airline has thebetter on-time performance?Interpret your results.

In Section 1.1, the collection of all possible outcomes (the universal set) of a randomexperiment is denoted by S and is called the outcome space. Given an outcome spaceS, let A be a part of the collection of outcomes in S; that is, A C S. Then A iscalled an event. When the random experiment is performed and the outcome of theexperiment is in A, we say that event A has occurred.

18 Chapter 1 Probability

where h = N (A) is the number of ways A can occur and m = N ( S) is the number ofways S can occur. Exercise 1.2-19 considers this assignment of probability in a moretheoretical manner.

It should be emphasized that in order to assign the probability him to theevent A, we must assume that each of the outcomes el, e2,. . . ,em has the sameprobability 1/ m. This assumption is then an important part of our probabilitymodel; if it is not realistic in an application, then the probability of the event Acannot be computed in this way. Actually, we have used this result in the simplecase given in Example 1.2-3 because it seemed realistic to assume that each ofthe possible outcomes in S = {HH,HT, TH, TT} had the same chance of beingobserved.

.~:f~'l'll:J.a.~:1 Let a card be drawn at random from an ordinary deck of 52 playing cards.esample space S is the set of m = 52 different cards, and it is reasonable to assumethat each of these cards has the same probability of selection, 1/52. Accordingly, ifA is the set of outcomes that are kings, then P(A) = 4/52 = 1/13 because thereare h = 4 kings in the deck. That is, 1/13 is the probability of drawing a card thatis a king, provided that each of the 52 cards has the same probability of beingdrawn. .

In Example 1.2-6, the computations are very easy because there is no difficulty inthe determination of the appropriate values of hand m. However, instead of drawingonly one card, suppose that 13 are taken at random and without replacement. Thenwe can think of each possible 13-card hand as being an outcome in a sample space, andit is reasonable to assume that each of these outcomes has the same probability. Forexample, to use the preceding method to assign the probability of a hand consistingof seven spades and six hearts, we must be able to count the number h of all suchhands, as well as the number m of possible 13-card hands. In these more complicatedsituations, we need better methods of determining hand m. We discuss some of thesecounting techniques in Section 1.3.

EXERCISES

1.2-L Of a group of patients having injuries, 28% A = {x:xisajack,queen,orking},visit both a physical therapist and a chiroprac- B = {x: x is a 9,10, or jack and x is red},tor and 8% visit neither. Say that the probabil- C - { .' I b}. f . .. h . aI h . ed th - x.xlSacu ,Ity 0 vlSltmg a p yslc t eraplst exce s e . .probability of visiting a chiropractor by 16%. D = {x: x IS a dIamond, a heart, or a spade}.What is the pr?bability ,!f. ? randomly selected Find (a) P(A), (b) P(A n B), (c) P(A U B),~s~on from thIS group Vlsltmg a physIcal thera- (d) P(C U D), and (e) P(C n D).p : .. 1.2-4. A coin is tossed four times, and the sequence of

1.2-2. An msurance company looks at. Its auto lDSurance heads and tails is observed.customers and finds that (a) alllDSure at least onecar, (b) 85% insure more than one car, (c) 23% (a) List each of the 16 sequences in the sampleinsure a sports car, and (d) 17% insure more than space S.one car, including a sports car. Find the probability (b) Let events A, B, C, and D be giventhat a customer selected at random insures exactly by A = {at least 3 heads}, B = { atone car and it is not a sports car. most 2 heads}, C = {heads on the third

1.2-3. Draw one card at random from a standard deck of toss}, and D = {1 head and 3 tails}.cards. The sample space S is the collection of the If the probability set function assigns 1/1652 cards. Assume that the probability set function to each outcome in the sample space,assigns 1/52 to each of the 52 outcomes. Let find (i)P(A), (ii)P(A n B), (iii)P(B),

J

.~

I

A = {x: x is a jack, queen, or king},B = {x: x is a 9,10, or jack and x is red},C = {x:xisaclub},D = {x:xis a diamond,aheart, or a spade}.

Find (a) P(A), (b) P(A n B), (c) P(A U B),(d)P(C U D), and (e) P(C n D).

1.2-4. A coin is tossed four times, and the sequence ofheads and tails is observed.

~

I

~

~

It

l.

(iv) P(A n C}, (v) P(D}, (vi) P(A U C}, 1.2-14. Let x equal a number that is selected randomlyand (vii) P(B n D}. from the closed interval from zero to one, [0,1].

1.2-5. A field of beans is planted with three seeds Use your intuition to assign values toper hill. F~r each hill ?f be~s, let Ai be the (a) P({x:O:S x:s 1/3}}.event that I seeds germInate, I = 0,1,2,3. Sup- (b) P({ .1/3:S S I}}pose that P(Ao} = 1/64, P(At} = 9/64, and x. x .P(A2} = 27/64. Give the value ofP(AJ}. (c) P({x:x = 1/3}}.

L2-6. Consider the trial on which a 3 is first observed (d) P({x: 1/2 < x < S}}.in successive rolls of a six-sided die. Let A be 1.2-15. A typical roulette wheel used in a casino hasthe event that 3 is observed on the first trial. 38 slots that are numbered 1,2,3 ,36,0,00,Let B be the event that at least two trials are respectively. The 0 and 00 slots are colored green.required to observe a 3. Assuming that each side Half of the remaining slots are red and half arehas probability 1/6, find (a) P(A}, (b) P(B}, and black. Also, half of the integers between 1 and(c) P(A U B}. 36 inclusive are odd, half are even, and 0 and 00

L2-7. A fair eight-sided die is rolled once. Let A = are defined to be neither odd nor even. A ball{2,4,6,8}, B = {3,6}, C = {2,S,7}, and D = is rolled around the wheel and ends up in one{1,3,S,7}. Assume that each face has the same of the slots; we assume that each slot has equalprobability. probability of 1/38, and we are interested in the

( ) (..) (B) number of the slot into which the ball falls.(a) Give the values of (i) P A, uP,(iii) P( C}, and (iv) P(D}. (a) Define the sample space S.

(b) Givethevaluesof(i)P(A n B}, (b) Let A = {O,OO}.GivethevalueofP(A}.(ii)P(B n C},and(iii)P(C n D). (c) Let B = {14,lS,17,18}. Give the value of

(c) Givethevaluesof(i}P(A U B}. P(B}.(ii)P(B U C), and (iii) P(C U D}, using (d) Let D = {x: xis odd}. Give the value ofTheorem 1.2-S. P(D}.

1.2-8. If P(A} = 0.4, P(B} = O.S, and P(A ~, B~n~ L2-16. The five numbers 1, 2, 3, 4, and S are writ-0.3, fin~ (a) P,(A U B}, (b) P(A n }, ten respectively on five disks of the same size(c~ P(A U B ). - , - and placed in a hat. Two disks are drawn with-

L2-9. GIven that P( A U B} - 0.76 and P( A U B } - out replacement from the hat, and the numbers0.87, find P( A}. written on them are observed.

1.2-10. During a visit to a primary care physician's office,the probability of having neither lab work nor (a) List the 10 possible outcomes of this experi-referral to a specialist is 0.21. Of those coming ment as unordered pairs of numbers.to that office, the probability of having lab work (b) If each of the 10 outcomes has probabilityis 0.41 and the probability of having a referral is 1/10, assign a value to the probability that0.S3. What is the probability of having both lab the sum of the two numbers drawn is (i) 3;work and a referral? (ii) between 6 and 8 inclusive.

1.2-1L Roll a fair six-sided die three times. Let At = . .. .b I .

{I 2 h fi II} A - {3 4 th L2-17. DIVIde a line segment mto two parts y se ectmgor on t e rst ro , 2 - or on e . d U .. . .d ll} dA - {S 6 th third ll} a pomt at ran om. se your IntuItIon to assignsecon ro , an J - or on e ro. b b.l . h th t h II .. h P(A . } = 1/3 . = 1 2 3 . a pro a I Ity to t e event ate onger seg-t IS gIven t at I , I , , , . I t t . I th th h rtP(Ai n A I.} = (1/3}2,i*j;andP(At n A2 n mentlstat eas wotlmes onger an es 0 er

segmen .AJ) = (1/3}J.1.2-18. Let the interval [-r, r] be the base of a semicircle.

(a) Use Theorem 1.2-6 to find P(At U A2 U If a point is selected at random from this interval,AJ ). assign a probability to the e~ent that the length

(b) Show that P(At U A2 U AJ} = 1 - (1 - of the perpendicular segment from the point to

1/3 }J. the semicircle is less than r/2.1.2-U. Prove Theorem 1.2-6. L2-19. Let S = At U A2 U ... U Am, where1.2-13. For each positive integer n, let P( {n}) = (1/2)". eventsAt, A2,..., Am are mutually exclusive and

Consider the events A = {n: 1 :S n :S 10}, exhaustive.B = {n: 1:Sn:S20},andC = {n:11:sn:S20}.Find (a) P(A}, (b) P(B}, (c) P(A U B}, (a) If P(At} = P(A2} = ... = P(Am}, show(d}P(A n B),(e}P(C},and(f}P(B'}. thatP(Ai} = l/m, i = 1,2,...,m.

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space,P(B),

Exercises 19

(a) Define the sample space S.(b) Let A = {O,OO}. Give the value ofP(A).(c) Let B = {14.15.17.18}. Give the value of

P(B).(d) Let D = {x : x is odd}. Give the value of

P(D).The five numbers 1, 2, 3, 4, and 5 are writ-ten respectively on five disks of the same sizeand placed in a hat. Two disks are drawn with-out replacement from the hat, and the numberswritten on them are observed.

(a) List the 10 possible outcomes of this experi-ment as unordered pairs of numbers.

(b) If each of the 10 outcomes has probability1/10, assign a value to the probability that

1.2-16.

the sum of the two numbers drawn is (i) 3;(ii) between 6 and 8 inclusive.

1.2-17. Divide a line segment into two parts by selectinga point at random. Use your intuition to assigna probability to the event that the longer seg-ment is at least two times longer than the shortersegment.

1.2-18. Let the interval [-r, r] be the base of a semicircle.If a point is selected at random from this interval,assign a probability to the e~ent that the lengthof the perpendicular segment from the point tothe semicircle is less than r12.

1.2-19. Let S = At U Az U ... U Am, whereeventsAl, Az,..., Am are mutuaUyexclusive andexhaustive.

(a) If P(Al) = P(Az) = ... = P(Am), showthatP(Aj) = 11m, i = 1,2,...,m.