15.6: Chapter 7
- Page ID
- 129936
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Your Turn
7.1
1. \(4 \times 15 = 60\)
7.2
1. \(4 \times 15 \times 3 \times 2 = 360\)
7.3
1. \(3 \times 10 \times 10 \times 10 \times 10 \times 10 \times 26 \times 26 \times 26 = {\text{5,272,800,000}}\)
7.4
1. 120
7.5
1. 720
2. 132
3. 70
7.6
1. 30
2. 24,024
3. 5,814
7.7
1. \(_{15}{P_3} = 2{,}730\)
7.8
1. combination
2. combination
7.9
1. 15
2. 45
3. 2,002
7.10
1. 30,856
2. 111,930
7.11
1. 290,004
7.12
1. \(\{\heartsuit ,\spadesuit ,\clubsuit ,\diamondsuit\}\)
2. {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}
3. {1, 2, 3, 4}
4. {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
7.13
1. Independent
2. Dependent
7.14
1. {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
7.15
1. {J\(\heartsuit\) Q\(\heartsuit\), J\(\heartsuit\) K\(\heartsuit\), Q\(\heartsuit\) J\(\heartsuit\), Q\(\heartsuit\) K\(\heartsuit\), K\(\heartsuit\) J\(\heartsuit\), K\(\heartsuit\) Q\(\heartsuit\)}
7.16
1. \(P(\text{H} < 5) = 1\)
2. \(0 < P(\text{H} < 4) < 1\)
3. \(P(\text{H} ≥ 5) = 0\)
7.17
1. \(\frac{1}{6}\)
2. \(\frac{2}{3}\)
3. \(\frac{1}{2}\)
7.18
1. \(\frac{5}16\)
7.19
1. \(\frac{1}{3}\)
2. \(\frac{1}{2}\)
3. \(\frac{2}{3}\)
7.20
1. \(\frac13
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= 1.3\% \)Callstack:
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7.21
1. theoretical
2. subjective
3. empirical
7.22
1. \(\frac1316\)
7.23
1. \(\frac14443680 = \frac{3}910\)
7.24
1. \(1 - \frac
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\approx 10.7\%\)Callstack:
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2. \(\frac
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\approx 0.86\%\)Callstack:
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7.25
1. \(13{:}3\)
2. \(12{:}4 = 3{:}1\)
7.26
1. The odds for \(E\) are \(4{:}1\) and the odds against \(E\) are \(1{:}4\).
7.27
1. \(P(E) = \frac{1}16\)
2. \(P(E) = \frac2.53.5 \approx 0.714\)
7.28
1. Mutually exclusive
2. Not mutually exclusive
3. Mutually exclusive
7.29
1. \(\frac{4}10 + \frac{2}10 = \frac{3}{5}\)
2. Not appropriate; the events are not mutually exclusive.
3. \(\frac{3}10 + \frac{2}10 = \frac{1}{2}\)
7.30
1. \(\frac{1}{2}\)
2. \(\frac{1}{2}\)
3. \(\frac{2}{3}\)
7.31
1. \(\frac{1}{3}\)
2. \(1\)
3. \(\frac{2}{3}\)
7.32
1. \(\frac{3}28\)
2. \(\frac{3}56\)
3. \(\frac{3}28\)
7.33
1.
| Roll | Probability |
|---|---|
| 1 | \(\frac{1}{3}\) |
| 2 | \(\frac{1}{4}\) |
| 3 | \(\frac{1}{6}\) |
| 4 | \(\frac{1}12\) |
| 5 | \(\frac{1}12\) |
| 6 | \(\frac{1}12\) |
7.34
1. Not binomial (more than two outcomes)
2. Not binomial (not independent)
3. Binomial
4. Not binomial (number of trials isn’t fixed)
7.35
1. 0.146
2. 0.190
3. 0.060
7.36
1. 0.2972
2. 0.6615
3. 0.0919
4. 0.5207
5. 0.3643
7.37
1. \(\frac10{3}\)
2. \(\frac{3}{2}\)
3. \(\frac25{4}\) = $6.25
7.38
1. If you roll the special die many times, the mean of the numbers showing will be around 3.33.
2. If you repeat the coin-flipping experiment many times, the mean of the number of heads you get will be around 1.5.
3. If you play this game many times, the mean of your winnings will be around $10.
7.39
1. If the player bets on 7, the expected value is \(- \$ 0.17\).
2. If the player bets on 12, the expected value is \(- \$ 0.14\).
3. If the player bets on any craps, the expected value is \(- \$ 0.11\).
The best bet for the player is any craps; the best bet for the casino is the bet on 7.
The best bet for the player is any craps; the best bet for the casino is the bet on 7.
Check Your Understanding
1. 540
2. 14
3. 1,024
4. 800
5. 25,920
6. 120
7. 120
8. 1,320
9. 1,680
10. \(_{15}{P_4} = 32{,}760\)
11. permutations
12. combinations
13. 66
14. 560
15. 20
16. 560
17. {0, 1, 2, 3, 4, 5, 6}
18.
| Crinkle Fries | Curly Fries | Onion Rings | |
|---|---|---|---|
| 8 Nuggets | 8 nuggets with crinkle fries | 8 nuggets with curly fries | 8 nuggets with onion rings |
| 12 Nuggets | 12 nuggets with crinkle fries | 12 nuggets with curly fries | 12 nuggets with onion rings |
19. 
20. {8 nuggets with crinkle fries, 8 nuggets with curly fries, 8 nuggets with onion rings, 12 nuggets with crinkle fries, 12 nuggets with curly fries, 12 nuggets with onion rings}
21. {history with Anderson, history with Burr, English with Carter, sociology with Johnson, sociology with Kirk, sociology with Lambert}
22. \(\frac{1}{4}\)
23. \(\frac{1}{2}\)
24. \(\frac{1}{2}\)
25. 0
26. theoretically
27. subjectively
28. empirically
29. \({\text{number of heads}} > 20\)
30. 89.9%
31. \(\frac
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\approx 0.278\%\)Callstack:
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32. \(\frac
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\approx 0.079\%\)Callstack:
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33. \(\frac
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\approx 0.16\%\)Callstack:
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34. \(\frac
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\approx 1.67\%\)Callstack:
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35. \(\frac
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\approx 1.9\%\)Callstack:
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36. \(\frac
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\approx 1.9\%\)Callstack:
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37. \(1{:}2\)
38. \(5{:}1\)
39. \(1{:}1\)
40. \(3{:}5\)
41. \(11{:}2\)
42. \(\frac{4}13\)
43. \(\frac{5}12\)
44. \(\frac{3}{5}\)
45. \(\frac{3}{5}\)
46. \(\frac{3}{5}\)
47. \(\frac{2}{5}\)
48. \(\frac{2}{5}\)
49. \(\frac{3}10\)
50. \(\frac{1}{4}\)
51. \(\frac{1}{9}\)
52. \(\frac{1}{6}\)
53. \(\frac{2}15\)
54. \(\frac{4}45\)
55. \(\frac{8}45\)
56. No (more than two outcomes)
57. Yes
58. No (number of trials is not fixed)
59. 0.9453
60. 0.1366
61. 0.8230
62. 4.1
63. If you roll this die many times, the mean of the numbers rolled will be around 4.1.
64. $0.417
65. If you play this game many times, the mean of the amount won/lost each time will be about 42 cents.
66. Yes; the expected value is positive.