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1.47. (a) 5 > 26 (b) 5 > 36 (c) 0.98 (d) 2 > 9 (e) 7 > 8 1.48. (a) Probability of king on first draw and no king on second draw. (b) Probability of either a king on first draw or a king on second draw or both. (c) No king on first draw or no king on second draw or both (no king on first and second draws). (d) No king on first, second, and third draws. (e) Probability of either king on first draw and king on second draw or no king on second draw and king on third draw. 1.49. (a) 1 > 3 (b) 3 > 5 (c) 11 > 15 (d) 2 > 5 (e) 4 > 5 1.50. (a) 4 > 25 (c) 16 > 25 (e) 11 > 15 (b) 4 > 75 (d) 64 > 225 (f) 1 > 5 1.51. (a) 29 > 185 (b) 2 > 37 (c) 118 > 185 (d) 52 > 185 (g) 104 > 225 (h) 221 > 225 (i) 6 > 25 ( j) 52 > 225 (i) 9 > 37 ( j) 26 > 111
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(g) 86 > 185 (h) 182 > 185
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1.52. (a) 3 > 10 (b) 1 > 10 (c) 3 > 5 1.54. 1 > 3 1.55. 21 > 56
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1.56. 21 > 31
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CHAPTER 1 Basic Probability
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1.60. (a) 13 1.63. n 5
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13 (b) 13
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1.61. 8
1.62. (a) 12 (b) 2520 (c) 720 1.66. (a) 8400 (b) 2520
1.64. 60
1.65. (a) 5040 (b) 720 (c) 240 1.68. 26 1.71. n 6
1.67. (a) 32,805 (b) 11,664 1.70. (a) 10 (b) 70 (c) 45 1.74. (a) 42,000 (b) 7000 1.77. (a) 17 (b) 163 1.79. (a) x 6 (b) x 4 (c) x 5 (d) x 8 1.80. 2016 6x 5 y 4x 3 y 5x 3 8x 6
1.69. (a) 120 (b) 72 (c) 12 1.72. 210 1.73. 840 1.76. (a) 150 (b) 45 (c) 100
1.75. (a) 120 (b) 2520
1.78. (a) 20 (b) 330 (c) 14 > 99 15x 4 y 2 6x 2y 2 20x 3 y 3 4xy3 y4 5x 3 16 x 5 15x 2 y 3 6xy 5 y6
10x 24x 4
10x 1 32x 2
1.81. (a) 5 > 18 (b) 11 > 36 (c) 1 > 36
1.82. (a) 47 > 52 (b) 16 > 221 (c) 15 > 34 (d) 13 > 17 (e) 210 > 221 (f) 10 > 13 (g) 40 > 51 (h) 77 > 442 1.83. 5 > 18 1.84. (a) 81 : 44 (b) 21 : 4 1.86. (6C3)(8C2) > 14C5
4C3) > 52C3
1.85. (a) (13C7)(13C2)(13C3)(13C1) > 52C13 (b) 4 > 52C13 1.87. (a) 91 > 216 (b) at least 17 1.89. 4(13C9)(39C4) > 52C13
1.88. (a) 4 13C3>/52C3 (b) (4C2 48C1 3
1.90. 211
1.91. (a) 120 (b) 60 (c) 72
CHAPTER 1 Basic Probability
1.92. (a) 63 > 125 (b) 963 > 1000 1.93. 1,419,857 > 34,012,224 (c) 45 (13C5) > 52C5 1.94. (a) 13 (b) 53
1.95. (a) 4 > 52C5 (b) (13)(2)(4)(6) > 52C5 1.96. 2 > 243 1.97. (a) 126 (b) n
(d) (5)(4)(3)(2) > (52)(51)(50)(49)
r 1Cn 1
1.98. (a) 462 (b) r
1.99. (a) 3 > 32 (b) 1 > 16 (c) 1 > 32 (d) 1 > 8 1.100. prob. A wins 61 > 216, prob. B wins 5 > 36, prob. of tie 125 > 216
1.101. (a) 12 > (52C13)(39C13) (b) 24 > (52C13)(39C13)(26C13)
CHAPTER 12 CHAPTER 2
Random Variables and Probability Distributions
Random Variables
Suppose that to each point of a sample space we assign a number. We then have a function defined on the sample space. This function is called a random variable (or stochastic variable) or more precisely a random function (stochastic function). It is usually denoted by a capital letter such as X or Y. In general, a random variable has some specified physical, geometrical, or other significance.
EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is S {HH, HT, TH, TT}. Let X represent the number of heads that can come up. With each sample point we can associate a number for X as shown in Table 2-1. Thus, for example, in the case of HH (i.e., 2 heads), X 2 while for TH (1 head), X 1. It follows that X is a random variable.