3.1 observing a single outcome

3.3 collection of all sample points

3.5 (1) probability falls between 0 and 1; (2) sum of probabilities of sample points equals 1

3.7 sum probabilities of sample points in A

3.9

1. .5

2. .3

3. .6

3.11 $P(A)=.55;P(B)=.50;P(C)=.70$

3.13

1. 10

2. 20

3. 15,504

3.15

1. (R₁R₂), (R₁R₃), (R₂R₃), (R₁B₁), (R₁B₂), (R₂B₁), (R₂B₂), (R₃B₁), (R₃B₂), (B₁B₂)

2. 1/10 for each sample point

3. $P(A)=1/10,P(B)=6/10,P(C)=3/10$

3.17

1. legs only, wheels only, both legs and wheels, neither legs nor wheels

2. $P(\text{Legs}\text{only})=.594,P(\text{Wheels}\text{only})=.189,P(\text{Both})=.075,P(\text{Neither})=.142$

3. .264

4. .669

3.19

1. Blue, Orange, Green, Yellow, Brown, and Red

2. $P(\mathrm{B}\mathrm{l}\mathrm{u}\mathrm{e})=.24$, $P(\mathrm{O}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e})=.20$, $P(\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n})=.16$, $P(\mathrm{Y}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w})=.14$, $P(\mathrm{B}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{n})=.13$, $P(\mathrm{R}\mathrm{e}\mathrm{d})=.13$

3. .13

4. .43

5. .76

3.21

1. .01

2. yes

3.23.615; .943

3.25.389

3.27

1. .091

2. .273

3.29

1. 6

2. .282, .065, .339, .032, .008, .274

3. .686

3.31

1. 28

2. 1/28

3.33 $693/1,686,366=.000411$

3.35

1. 15

2. 20

3. 15

4. 6

5. 64

3.37 no, probabilities are equal

3.39 if either A or B or both occur

3.41 all sample points that are not in A

3.43 $P(A\cup B)=P(A)+P(B)$

3.45

1. b. $P(A)=7/8,P(B)=1/2,P(A\cup B)=7/8,P(A^c)=1/8,P(A\cap B)=1/2$

2. c. 7/8

3. d. no

3.47

1. ${\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

2. 13/20

3. 1

4. 2/5

5. ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

6. 7/20

7. 1

8. 1/4

3.49

1. $A:\{(1,6)$, $(2,5)$, $(3,4)$, $(4,3)$, $(5,2)$, $(6,1)\};B:\{(1,4)$, $(2,4)$, $(3,4)$, $(4,4)$, $(5,4)$, $(6,4)$, $(4,1)$, $(4,2)$, $(4,3)$, $(4,5),(4,6)\};A\cap B:\{(3,4)$, $(4,3)\};A\cup B:\{(1,4)$, $(2,4)$, $(3,4)$, $(4,4)$, $(5,4)$, $(6,4)$, $(4,1)$, $(4,2)$, $(4,3)$, $(4,5)$, $(4,6)$, $(1,6)$, $(2,5)$, $(5,2)$, $(6,1)\};A^c:\{(1,1)$, $(1,2)$, $(1,3)$, $(1,4)$, $(1,5)$, $(2,1)$, $(2,2)$, $(2,3)$, $(2,4)$, $(2,6)$, $(3,1)$, $(3,2)$, $(3,3)$, $(3,5)$, $(3,6)$, $(4,1)$, $(4,2)$, $(4,4)$, $(4,5)$, $(4,6)$, $(5,1)$, $(5,3)$, $(5,4)$, $(5,5)$, $(5,6)$, $(6,2)$, $(6,3)$, $(6,4)$, $(6,5)$, $(6,6)\}$

2. $P(A)=1/6,P(B)=11/36,P(A\cap B)=1/18,P(A\cup B)=5/12,P(A^c)=5/6$

3. 5/12

4. no

3.51

1. .65

2. .72

3. .25

4. .08

5. .35

6. .72

7. .65

8. A and C, B and C, C and D

3.53.858

3.55

1. b. .43

2. c. .57

3.57

1. $\{\text{Non-REM/Non-REM}}$, $\{\text{Non-REM/REM}}$, $\{\text{Non-REM/Wake}}$, $\{\text{REM}/\text{Non-REM}}$, $\{\text{REM/REM}}$, $\{\text{REM/Wake}}$, $\{\text{Wake/Non-REM}}$, $\{\text{Wake/REM}}$, $\{\text{Wake/Wake}}$

2. .6385, .0032, .0347, .0069, .1524, .0035, .0318, .0072, .1218

3. .1628

4. .16

5. .0032

3.59

1. .333

2. .351

3. .263

4. .421

3.61

1. .156

2. .617

3. .210

4. .449

5. yes

3.63

1. .19

2. .405

3. .275

3.65 $7/45=.156$

3.69

1. $P(A\cap B)=P(A)P(B)$

2. $P(A\cap B)=P(B|A)P(A)=P(A|B)P(B)$

3.71

1. false

2. true

3. false

3.73

1. .5

2. .25

3. no

3.75

1. .08

2. .40

3. .52

3.77

1. $P(A)=.4;P(B)=.8;P(A\cap B)=.3$

2. $P(E_1|A)=.25,$ $P(E_2\text{|}A)=.25,P(E_3|A)=.5$

3. .75

3.79 no

3.81

1. .75

2. .30

3.83 .3195

3.85

1. .26

2. .013

3.87 .40

3.89 .013; independent events

3.91 .35

3.93

1. .23

2. .729

3.95

1. $38/132=.2879$

2. $29/123=.236$

3.97

1. .406

2. .294

3.99 $P(A|B)=0$

3.101

1. .1

2. .522

3. The person is guessing

3.103. $5\beta +\alpha \beta -\alpha {\beta }^2$

3.105

1. $A\cup B$

2. B^c

3. $A\cap B$

4. $A^c|B$

3.107

1. 0

2. no

3.109 .5

3.111

1. c. $P(A)=1/4;P(B)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

2. e. $P(A^c)=3/4;P(B^c)=1/2;P(A\cap B)=1/4;P(A\cup B)=1/2;P(A|B)=1/2;P(B|A)=1$

3. f. no, no

3.113

1. no

2. .3, .1

3. .37

3.115

1. (None), (1 or 2), (3–5), (6–9), (10 or more)

2. $P(\mathrm{N}\mathrm{o}\mathrm{n}\mathrm{e})=.25$, $P(1\mathrm{ o}\mathrm{r }2)=.31$, $P(3-5)=.25$, $P(6-9)=.05$, $P(10\mathrm{ o}\mathrm{r }\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{e})=.14$

3. .44

3.117.05

3.119

1. false

2. true

3. true

4. false

3.121

1. .261

2. Trunk–.85, Leaves–.10, Branch–.05

3.123

1. $\{\text{Single},\text{shore}\text{parallel};\text{Other};\text{Planar}\}$

2. .1/3, 1/3, 1/3

3. 2/3

4. $\{\text{No dunes/flat}$; $\text{Bluff/scarp}$; $\text{Single}\text{dune}$; $\mathrm{N}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{d}\}$

5. 1/3, 1/6, 1/3, 1/6

6. 2/3

3.125

1. School laboratory, In Transit, Chemical plant, Non-chemical plant, Other

2. .06, .26, .21, .35, .12

3. .06

4. .56

5. .74

6. .000216

7. no

3.127

1. $\{11,13,15,17,29,31,33,35\}$

2. $\{2,4,6,8,10,11,13,15,17,20,22,24,26,28,29,31,$ $33,35,1,3,5,7,9,19,21,23,25,27\}$

3. $P(A)=9/19$, $P(B)=9/19$, $P(A\cap B)=4/19$, $P(A\cup B)=14/19$, $P(C)=9/19$

4. $\{11,13,15,17\}$

5. 14/19,

6. no 2/19

7. $\{1,2,3,\dots ,29,31,33,35\}$

8. 16/19

3.129

1. .157

2. .029

3. .291

4. .183

3.131

1. .64, .32, .04

2. .72, .22, .06

3. dependent

3.133

1. .116

2. .728

3.135

1. .7127

2. .2873

3. .9639

4. .3078

5. .0361

6. at least 3

3.137.748

3.139

1. 1 to 2

2. .5

3. .4

3.141

1. .25

2. .0156

3. .4219

4. .001, .000000001, .9970

3.143.993

3.145 Marilyn