8.3: Probability
- Page ID
- 192932
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sample Spaces and Probability
1) {1, 2, 3, 4, 5, 6}
3) {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}
5) {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
7) 4/52 = 2/26
9) 13/52
11) 16/52 = 4/13
13) 6/20 = 3/10
15) 13/20
17) 3/8
19) 6/8 = 3/4
21) 2/8 = 1/4
23) 4/36 = 1/9
25) 7/36
27) 6/36 = 1/6
29) 8/12 = 2/3
31) 6/12 = 1/2
33) 1/6
Mutually Exclusive Events and the Addition Rule
1) Yes
3) No
5) No
7) 16/52 = 4/13
9) 13/36
11) 40%
13) 30/100 = 3/10
15) 94/100 = 47/50
17) 68/100 = 17/25
19) Not mutually exclusive
21) 0
23) 0.5
25) 0.4
Probability Using Tree Diagrams and Combinations
1) 20/56 = 5/14
3) 6/65
5) 1/6
7) 3/10
9) 10/220 = 1/22
11) 56/220 = 14/55
13) 45/1365 = 3/91
15) 21/1365 = 1/65
17) 324/1365 = 108/455
19) 79,092/2,598,960 = 507/16,660 \(\approx\) 0.0304
21) 24/2,598,960 = 1/108,290 \(\approx\) 0.000009234
23) 1,584/2,598,960 = 33/54145 \(\approx\) 0.0006095
25) 1890/12376 = 135/884 \(\approx\) 0.1527
27) 1680/12376 = 30/221 \(\approx\) 0.1357
29) 180/792 = 5/22 = \(0.2\overline{27}\)
31) 21/792 = 7/264 = \(0.26\overline{51}\)
33) 0.972864
Conditional Probability
1) 4/12 = 1/3
3) 1/3
5) 30/44 = 15/22
7) 6/54 = 3/26
9) a) 1/6; b) 1/4
11) 0.12
13) 0.4
15) 0.696
17) 1/2
19) 3/4
21) 84/528 = 7/44
23) 84/2096 = 21/524
25) 188/796 = 47/199
Independent Events
1) 2/3
3) 50/150 = 1/3
5) a) 3/4; b) 1/2; c) 1/2; d) no
7) 0.12
9) 0.36
11) yes
13) a) 28/100 = 7/25; b) 82/100 = 41/50
15) a) 7/8; b) 3/4; c) 3/4; d) no
17) a) yes; b) 0.175
19) a) no; b) 0.081
Bayes' Formula
1) a) 0.6458; b) 0.4706; c) 0.625
3) The Republican party
5) 0.7787
7) a) 0.045; b) 0.2667; c) 0.03
9) a) 0.02; b) 0.111
Chapter Review
1) 3/36 = 1/12; 4/36 = 1/9
2) 8/12 = 2/3; 7/12
3) 8/52 = 2/13; 16/52 = 4/13
4) 3/5; 9/10
5) 3/20; 1/20; 5/6; 1/6
6) 1/16; 1/4
7) 3/4; 0
8) 0.72
9) 40%
10) independent
11) 3/4; 0.45
12) 0.8144
13) 0.22
14) 0.45278
15) a) 111540/2598960 = 143/3332 \(\approx\) 0.0429; b) 949104/2598960 = 1521/4165 \(\approx\) 0.3652; c) 1349088/2598960 = 2162/4165 \(\approx\) 0.5191; d) 36/2598960 = 3/216580 \(\approx\) 0.00001385
16) 9/20; 10/27; 15/33; 11/20; no; yes
17) 0.4
18) 3/14; 37/42; 2/7; 35/84
19) no
20) 0
21) 0.65
22) 0.36
23) 5/6
24) 0.2
25) 0.5
26) 0.3