FirstonProbability

FirstonProbability - Basic Probability Whenever we observe...

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Unformatted text preview: Basic Probability Whenever we observe something, we see distribution of outcomes. Events occur with randomness. We can look for patterns in the randomness to allow us to see through it to underlying relationships. Probability is the set of ideas that allows us to understand random events. We will begin with gambling devices because they are designed to produce controlled randomness yet keep the pattern predictable. Flip coins Roll dice Draw balls from an urn Draw cards from a deck Bet on a horse race On Wednesday, we will develop axioms to generalize what we see today in individual examples. The basic idea is count all of the equally likely outcomes in an outcome of interest as a ratio to count of all possible equally likely outco th the to us to use they mness yet s to generalize ly likely ly likely outcomes. Flip A Coin Can a spreadsheet flip a coin? Fig 5_2 Seeing Coin Flips Enter any number and touch return to recalculate: Here is the result of 100 flips of a coin. Heads Tails Count 52 48 Percentage 0.52 0.48 Forecast by Theory 0.50 0.50 difference -0.02 0.02 One Hundred Coin Flips 0.70 0.50 0.45 Percentage 0.60 0.50 0.40 0.52 0.48 0.35 0.40 0.30 0.30 0.20 0.25 0.15 0.20 0.10 0.10 0.00 0.05 Heads Tails 0.00 How does a spreadsheet flip a coin? We use the = RAND() function to generate a random number between zero and one. Rounding to a whole number gives a value equally likely to be zero as one. Call a one a head. (See cell U6 for the full simulation.) Page 5 Fig 5_2 Enter any number and touch return to induce the spreadsheet t Here is the result of 100 flips of No Heads One Head Count of results 22 54 ### 0 1 Percentage 0.22 0.54 Forecast by Theory ### 0.25 0.5 difference 0.03 -0.04 Frequency & Probability 100 Flips of Two Coins 0.7 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.6 0.5 0.4 0.3 0.2 0.1 0 -1 0 1 2 Number of Heads Probability gives the expectation that 25 percent of the 100 tosses will yield two heads. 50 percent one head, and 25 percent no heads. Page 6 3 Fig 5_2 turn to recalculate: 2 0.50 0.45 0.40 0.35 0.30 0.25 0.20 Probability theory gives the expectation that 50 percent of the 100 tosses will yield heads. 0.15 0.10 0.05 0.00 Head Page 7 Fig 5_2 duce the spreadsheet to recalculate: 0 flips of two coins. 5 Two Heads 24 ### 2 0.24 0.25### 0.01 ns 2 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 3 s Page 8 Fig 5_2 Repetitions 4 5 1 0 0 1 1 2 1 0 1 0 1 5 4 3 2 1 3 2 1 1 1 22 54 24 0 1 0 1 individual 1 coin flips 2 3 0 1 0 0 0 4 1 1 1 1 1 0 1 0 1 0 3 2 1 1 1 1 1 0 0 0 5 4 2 2 1 2 2 1 1 0 0 1 0 1 0 0 0 0 1 0 1 0 3 5 counts of Count on one flip 52 count of events with two flips zero Number of Heads one two 100 Page 9 Fig 5_2 Page 10 Fig 5_2 6 1 0 0 0 1 7 8 0 0 1 0 1 1 1 1 1 1 9 10 11 12 13 14 15 1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 18 0 1 1 0 1 17 0 0 0 0 0 2 1 1 0 1 2 1 1 1 0 5 4 2 2 1 2 2 1 1 1 2 2 1 1 1 1 1 0 0 0 2 1 1 0 0 3 2 1 1 0 3 3 1 2 1 1 1 1 1 0 3 2 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 16 Page 11 20 0 0 1 1 1 19 1 0 1 0 1 22 1 1 0 0 0 21 0 0 0 0 1 0 0 0 0 0 3 2 2 2 0 1 0 0 0 0 1 24 1 1 0 1 0 23 1 0 0 1 1 3 2 1 1 1 2 2 0 0 1 1 0 1 0 0 3 3 1 1 1 3 2 2 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 Fig 5_2 Page 12 Fig 5_2 25 1 0 1 0 1 26 28 1 1 0 0 1 27 0 1 0 1 0 30 0 1 1 1 1 29 1 0 0 1 1 3 2 1 1 1 3 2 1 0 1 0 1 0 1 0 0 32 1 0 1 0 1 31 1 0 0 0 1 2 2 1 1 0 4 3 2 2 0 0 1 0 0 0 1 34 0 1 1 1 0 33 0 0 1 0 1 3 2 2 1 1 3 2 1 1 1 0 1 0 0 1 0 36 0 0 0 1 0 35 0 1 1 1 0 2 1 1 0 1 3 3 1 2 0 1 0 0 0 0 1 2 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 Page 13 38 0 1 1 0 0 37 1 1 1 0 0 40 0 1 1 0 0 39 0 1 0 1 1 0 1 0 0 0 41 1 1 0 1 0 3 3 1 2 0 2 2 0 1 0 3 3 0 1 1 2 2 0 1 0 3 2 2 1 0 1 1 0 0 0 3 3 1 1 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 Fig 5_2 Page 14 Fig 5_2 44 1 1 0 1 1 43 1 1 0 1 0 4 3 2 1 1 0 1 0 42 46 0 0 0 1 1 45 0 0 1 1 0 48 1 1 1 1 1 47 0 0 0 0 0 3 3 1 1 1 2 1 2 1 0 0 1 0 0 1 0 50 1 0 1 0 0 49 0 0 1 0 0 2 2 1 2 0 5 4 2 2 1 0 0 1 0 0 1 52 1 1 1 1 0 51 0 0 1 0 0 0 0 0 0 0 2 2 0 1 1 1 0 0 0 1 0 1 1 0 1 0 4 4 1 2 1 0 1 0 0 0 1 Page 15 54 0 0 1 1 0 53 0 1 1 1 1 56 1 1 0 1 1 55 0 1 0 0 0 1 1 0 1 0 2 2 1 2 0 0 1 0 0 0 1 58 1 1 1 0 0 57 1 0 1 0 1 4 3 2 2 0 4 3 2 1 1 1 1 0 0 0 3 3 0 1 1 3 2 1 1 1 2 2 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 Fig 5_2 Page 16 Fig 5_2 59 0 0 0 0 1 60 62 0 1 1 1 0 61 1 0 0 0 1 64 0 0 1 1 0 63 1 1 1 0 0 1 0 1 0 0 3 3 1 2 0 1 0 0 0 0 1 66 1 1 1 1 1 65 0 1 1 0 1 2 1 1 0 1 2 2 1 2 0 1 0 0 0 0 1 68 1 1 1 1 0 67 0 0 1 0 0 3 3 0 1 1 5 4 2 2 1 0 1 0 0 0 1 70 0 1 0 0 1 69 1 1 1 0 0 3 2 1 1 0 4 4 1 2 1 0 1 0 0 0 1 1 1 0 1 0 2 1 1 0 0 0 1 0 1 0 0 Page 17 72 0 1 0 0 1 71 1 0 1 0 0 74 1 0 1 0 1 73 1 0 0 1 0 1 1 0 1 0 75 1 1 1 1 1 3 3 0 1 1 2 1 1 0 0 2 2 0 1 1 3 2 1 1 1 2 2 1 1 1 3 3 1 1 1 5 4 2 2 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 Fig 5_2 Page 18 Fig 5_2 78 0 1 0 0 0 77 0 0 0 1 1 1 1 0 0 0 1 0 0 76 80 1 1 1 0 0 79 1 0 0 0 0 82 1 0 1 1 1 81 1 0 1 0 1 2 1 2 1 0 3 3 0 1 1 0 1 0 0 1 0 84 0 1 0 0 0 83 1 0 1 1 0 1 1 0 0 1 4 3 2 2 1 1 0 0 0 0 1 86 0 0 1 1 0 85 1 0 0 0 1 3 2 1 1 1 1 1 0 0 0 0 1 0 1 0 0 3 3 1 2 1 2 2 1 2 0 0 0 1 0 0 1 Page 19 88 0 1 1 1 1 87 0 0 1 0 0 90 1 1 1 0 0 89 0 1 1 0 0 2 1 1 0 1 4 3 2 2 0 1 0 0 0 0 1 92 1 1 1 1 1 91 1 0 0 1 0 1 1 0 1 0 3 3 0 1 1 2 2 0 1 0 5 4 2 2 1 2 2 1 1 1 3 3 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 0 Fig 5_2 Page 20 Fig 5_2 93 0 0 0 1 0 94 96 0 1 1 1 0 95 1 1 0 0 1 98 1 0 0 0 1 97 0 1 0 1 1 1 1 1 1 0 3 3 1 2 0 3 2 1 0 1 2 1 1 0 1 3 2 2 1 0 4 3 1 1 1 3 2 1 1 1 4 3 2 2 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 99 100 1 0 0 1 1 1 0 1 1 1 Page 21 Fig 5_2 Page 22 Fig 5_3 Two Dice The cells in the table show the sum of the numbers on the two faces of the dice. Second First 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 5 6 7 6 7 8 7 8 9 8 9 10 9 10 11 10 11 12 There are thirty-six possible outcomes from rolling two dice, each with six equally likely faces. The possible sum of the numbers on the two top faces are enumerated in the table Each cell is equally likely to occur. a. What is the probability of getting a sum that is 10 or larger? ten occurs in three ways, eleven in two, and twelve in one, so the cou The probability of getting a sum that is 10 or larger is 6/36. b. What is the probability of getting a sum that is even? The probability of getting a sum that is even is 18/36. c. What is the probability of getting an even sum or a sum that is 10 or more? To find the probability of getting a sum that is even or that is ten or m Page 23 Fig 5_3 the 14 events that are less than ten plus the six ten or more. The probability is then 20/36. d. If the first die shows a "5," what is the probability that the sum is 10 or more? If the first die shows a five, then only six cells are relevant. Of the six, two show a sum 10 or more. So, the probability is 2/6. Page 24 Fig 5_3 n one, so the count is 6. is 10 or more? r that is ten or more, count Page 25 Fig 5_3 r more. um is 10 or more? Page 26 Can a spreadsheet roll a die? Fig 5_4 Seeing Dice Enter any number in the yellow cell to the right to roll a pair of dice 200 time Roll Two Dice sum on faces count predicted by probability difference 2 9 5.6 3 17 11.1 4 15 16.7 5 25 22.2 3.4 5.9 -1.7 6 30 27.8 2.8 2.2 -1.3 Enter a number and to 200 Rolls of a Pair of Dice Frequency & Predicted count predicted 50 35.0 30.0 40 25.0 30 20.0 20 15.0 10.0 10 0 5.0 2 3 4 5 6 7 8 9 10 11 12 Sum of Faces Page 28 7 32 33.3 - Fig 5_4 Page 29 Fig 5_4 Page 30 Fig 5_4 Page 31 Fig 5_4 Page 32 Fig 5_4 Page 33 Fig 5_4 of dice 200 times. 8 23 27.8 9 24 22.2 10 15 16.7 11 6 11.1 12 4 5.6 sum 200 200 -4.8 1.8 -1.7 number and touch return: -5.1 4 -1.6 0.0 Page 34 Fig 5_4 Page 35 Fig 5_4 Page 36 Fig 5_4 Page 37 Fig 5_4 Page 38 Fig 5_4 Page 39 Fig 5_4 Total on Face Count of Each Counting Results Column AA to the right numbers each roll. Columns AB and AC contain the result for each die. Column AD sums the faces. Columns AF through AP contain If statements to count whether a face amount is of a given size. Each row contains the result of one roll. Page 40 Summing all the values in AF counts the roll # #1 #2 sum 1 5 5 10 2 3 6 9 3 1 6 7 4 2 5 7 5 2 6 8 6 3 3 6 7 3 1 4 8 5 4 9 9 2 3 5 10 1 6 7 11 6 1 7 12 4 3 7 13 1 3 4 14 1 2 3 15 3 6 9 16 6 1 7 17 5 3 8 18 6 5 11 19 3 2 5 20 3 4 7 21 1 5 6 22 6 3 9 23 6 5 11 24 1 2 3 25 1 1 2 26 1 1 2 27 2 4 6 28 4 2 6 29 6 4 10 30 4 3 7 31 4 3 7 for each die. Column AD sums the faces. Fig Columns AF through AP contain 5_4 If statements to count whether a face amount is of a given size. Each row contains the result of one roll. Summing all the values in AF counts the number of times the total is two. Thus sum is in AF2. Page 41 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 3 6 1 4 6 3 1 4 3 3 6 1 6 1 3 1 6 1 1 3 2 6 6 4 2 5 5 4 6 1 2 1 1 6 6 2 1 2 3 2 4 4 6 2 2 5 3 4 3 1 4 4 5 6 6 5 6 5 3 3 3 1 2 5 4 4 2 2 1 1 2 3 1 1 5 9 3 8 10 9 3 6 8 6 10 4 7 5 7 6 12 7 6 9 7 9 9 7 3 7 10 8 10 3 4 2 2 8 9 3 2 Fig 5_4 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 Page 42 4 4 3 2 1 1 4 2 2 6 3 3 1 5 3 2 6 1 5 5 2 3 5 3 6 2 1 5 5 4 3 1 5 5 4 1 3 1 2 6 2 6 4 6 4 3 2 6 3 2 2 6 6 6 3 3 5 6 5 4 1 2 3 4 2 4 1 6 3 2 3 4 5 3 5 6 9 4 7 5 10 6 5 8 9 6 3 7 9 8 12 4 8 10 8 8 9 4 8 5 5 7 9 5 9 4 7 8 8 6 6 Fig 5_4 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 Page 43 4 1 2 2 3 1 4 5 6 6 4 2 1 5 4 5 3 1 5 1 4 3 6 2 3 1 1 3 1 3 2 6 5 2 2 5 3 2 6 1 3 2 2 6 1 2 1 5 2 5 2 2 1 5 4 2 2 1 2 2 6 2 1 2 1 1 5 1 5 6 4 2 5 2 6 7 3 5 5 3 10 6 8 7 9 4 6 7 6 6 8 5 7 3 5 5 8 8 5 2 3 4 2 8 3 11 11 6 4 10 5 Fig 5_4 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 Page 44 6 4 2 4 1 4 5 4 3 1 4 2 2 2 2 2 6 4 6 3 5 3 4 4 5 2 3 2 6 6 1 3 1 1 2 6 4 2 2 1 2 4 3 4 1 6 2 5 6 3 1 4 2 6 2 1 6 6 3 6 1 4 3 3 5 3 2 2 6 1 6 4 6 2 8 6 3 6 5 7 9 5 9 3 9 8 5 3 6 4 12 6 7 9 11 6 10 5 9 5 6 7 9 8 3 9 2 7 6 12 6 Fig 5_4 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 Page 45 3 1 2 4 3 6 5 1 1 6 1 4 5 5 5 6 3 1 1 1 5 2 6 6 3 1 4 6 5 1 4 6 1 5 5 2 1 1 4 5 3 1 5 7 8 7 4 10 11 6 2 10 7 5 10 10 7 7 4 5 6 4 6 Fig 5_4 otal on Face 2 unt of Each 9 3 17 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 4 15 5 6 7 8 9 10 11 12 25 30 32 23 24 15 6 4 count sums equal 3 4 5 6 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 to 8 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 each value 9 10 11 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Page 46 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 200 Fig 5_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 Page 47 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Fig 5_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 Page 48 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Fig 5_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 Page 49 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Fig 5_4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Page 50 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 Fig 5_4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Page 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Fig 5_5 Balls in an Urn 15 solid blue 30 solid red 15 striped blue 40 striped red An urn contains one hundred balls, well stirred. When a ball is drawn, each ball is equally likely to be drawn. After a ball is drawn, it is replaced and the urn churned to assure that each ball is equally likely to be drawn on the next draw. Enter any number to create new random numbers. 4 Random draw from urn: solid blue Enter any number in the yellow box to generate a random draw from the urn. The result appears immediately. (The simulation is built at cell AA1.) What is the probability of drawing a blue ball? What is the probability of drawing a striped ball? What is the probability of drawing a ball that is blue an What is the probability of drawing a ball that is either Page 52 Fig 5_5 Page 53 Fig 5_5 Page 54 Fig 5_5 ue ball? riped ball? all that is blue and striped? all that is either blue or striped? Page 55 Fig 5_5 Page 56 Fig 5_5 Page 57 Fig 5_5 Page 58 Fig 5_5 Page 59 Fig 5_5 Page 60 Fig 5_5 3 How Does a Spreadsheet Draw a Ball from an Urn? Columns AB to AD represent the 100 balls, each with two attributes. Column AB labels them from 1 to 100. Cell AA1 contains a random whole number between 1 and 100. When the spreadsheet recalculates, it computes a new random number. The values of "striped" and "red" are pulled from the randomly selected row using Excel's VLOOKUP function. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Page 61 Fig 5_5 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 Page 62 Fig 5_5 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Page 63 Fig 5_5 blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue blue red solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped solid red solid red solid red solid red solid red solid red solid red red solid solid Page 64 Fig 5_5 red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red red solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid solid striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped Page 65 Fig 5_5 red red red red red red red red red red red red red red red red red red striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped striped Page 66 Fig 5_6 Seeing the Urn Probability & Relative Frequency Enter number of balls of each type that are in the urn. 100 Draws Enter Probability Frequency rel freq blue striped 20 ### ### 24% 25 25% blue plain 23 ### ### 28% 25 25% green striped 25 ### ### 30% 35 35% green plain 15 ### ### 18% 15 15% sum 83 100% 100 100% absolute % error/4 2.9% 100 Draws from the Urn 50% 40% 30% 20% 10% 0% blue striped blue plain green striped green plain 35% 30% 25% 20% 15% 10% 5% 0% Ball Type Column C Column E s & Relative Frequencies recalculate Enter number to draw again 3 1000 Draws with Replacement 50% 35% 30% 40% 25% 30% 20% 20% Page 67 15% 10% Enter numbers establish any f like. Enter a numbe how much vari the 100 draws. Probabilities & Relative Frequ 50% 35% 40% 30% Fig 5_6 25% 30% 20% 20% 15% 10% 10% 0% 5% blue striped blue plain green striped green plain The lower char 1000 draws. 0% probabilities pastel 0.18 dark 0.05 linen not white 0.60 Use the simulator to compare how a random s shoppers might yield differ pattern of results, g proportions in the population. Enter these values in the simulator and observe the patterns of samples d labels pastel dark white other Page 68 count 18 5 40 37 Fig 5_6 Page 69 Fig 5_6 Page 70 Fig 5_6 Page 71 Fig 5_6 Page 72 Fig 5_6 Page 73 Fig 5_6 Page 74 Fig 5_6 Page 75 Fig 5_6 Page 76 Fig 5_6 Page 77 Fig 5_6 Page 78 Fig 5_6 Page 79 Fig 5_6 Page 80 Fig 5_6 Page 81 Fig 5_6 Page 82 Fig 5_6 Page 83 Fig 5_6 Page 84 Fig 5_6 Page 85 Fig 5_6 Page 86 Fig 5_6 Page 87 Fig 5_6 Page 88 Fig 5_6 Page 89 Fig 5_6 Page 90 Fig 5_6 Page 91 Fig 5_6 Page 92 Fig 5_6 Page 93 Fig 5_6 Page 94 Fig 5_6 1000 Draws Frequency rel freq 261 26% 285 29% 284 28% 170 17% 1000 100% 1.4% The range at S8 de values between on based on the propo in the urn. Enter numbers in the yellow cells to establish any four-group pattern you like. The random numbe R are random draw 100 numbers so fal according to the ap probabilities. Enter a number several times to see how much variability occurs within the 100 draws. Columns S through statements to iden each draw. The "absolute % error/4" is the absolute value of the probability in each cell minus the relative frequency in the cell, summed over the four cells, and divided by 4 to state is as the per cell percentage error. The 1000 draws generally have much smaller absolute percentage errors than the 100 draws. Page 95 The row at S3 coun of balls in each bin 100 draws. The row the bin totals for th Fig 5_6 The lower chart shows the result of 1000 draws. abilities re how a random survey of pattern of results, given the n. in the simulator tterns of samples drawn. Page 96 Fig 5_6 Page 97 Fig 5_6 Page 98 Fig 5_6 Page 99 Fig 5_6 Page 100 Fig 5_6 Page 101 Fig 5_6 Page 102 Fig 5_6 Page 103 Fig 5_6 Page 104 Fig 5_6 Page 105 Fig 5_6 Page 106 Fig 5_6 Page 107 Fig 5_6 Page 108 Fig 5_6 Page 109 Fig 5_6 Page 110 Fig 5_6 Page 111 Fig 5_6 Page 112 Fig 5_6 Page 113 Fig 5_6 Page 114 Fig 5_6 Page 115 Fig 5_6 Page 116 Fig 5_6 Page 117 Fig 5_6 Page 118 Fig 5_6 Page 119 Fig 5_6 Page 120 Fig 5_6 Page 121 Fig 5_6 Page 122 Fig 5_6 One Random Draw 51 The range at S8 defines bin values between one and 100 based on the proportion of balls in the urn. The random numbers in column R are random draws from the 100 numbers so fall into the bins according to the appropriate probabilities. Columns S through V use IF statements to identify the bin for each draw. The row at S3 counts the number of balls in each bin over all the 100 draws. The row at S7 counts the bin totals for the 1000 draws. Cumulative 100 draws Cumulative 1000 draws Bin Values 25 25 50 25 261 261 24.1 546 285 51.81 to A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Page 123 64.579 0.136 5.359 33.106 10.389 11.503 18.613 66.873 12.604 7.264 25.841 26.565 89.015 3.308 97.411 60.097 75.906 23.356 56.548 92.449 67.981 69.512 7.561 33.053 78.490 83.532 45.977 to B 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 Fig 5_6 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Page 124 21.516 64.894 55.807 23.699 29.472 55.943 29.058 62.578 66.331 40.560 81.191 33.204 53.164 88.455 59.046 79.729 77.470 62.354 77.140 37.566 38.260 0.496 94.115 30.709 68.477 63.627 38.270 1.531 42.117 21.802 47.508 63.633 86.695 3.315 87.332 16.167 59.257 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 Fig 5_6 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 Page 125 16.389 78.745 25.589 56.950 59.936 58.793 10.114 48.391 17.839 89.843 25.861 80.193 66.983 63.428 18.453 67.480 57.542 49.162 35.957 21.169 87.432 37.488 63.286 9.233 84.996 26.919 95.928 88.311 14.250 12.095 47.568 30.640 90.841 73.156 87.589 50.777 31.949 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 0 1 1 Fig 5_6 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 Page 126 97.703 99.168 49.788 87.546 25.029 29.980 54.530 88.457 48.434 22.009 45.999 97.595 57.967 67.168 85.027 95.454 30.454 94.260 80.450 57.372 90.188 68.761 71.623 2.284 16.329 2.262 93.124 89.485 89.851 43.901 21.435 87.555 43.070 71.222 75.101 68.099 1.203 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 Fig 5_6 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 Page 127 29.631 56.556 49.636 51.641 2.555 47.232 9.607 69.723 32.259 5.062 0.177 26.519 85.512 57.549 16.707 54.273 29.172 18.991 70.601 31.434 12.115 60.087 21.285 56.017 81.521 8.840 99.086 52.744 83.942 67.185 53.947 13.573 23.741 3.583 65.214 26.296 50.815 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 Fig 5_6 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 Page 128 74.821 96.019 83.074 79.883 96.195 9.593 65.394 53.744 26.300 19.667 82.916 45.291 90.268 14.350 57.407 50.355 35.635 13.423 31.876 44.475 12.510 84.620 28.417 79.695 38.567 41.990 3.436 42.150 7.203 29.732 92.965 82.024 25.751 76.039 61.907 21.946 85.632 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 Fig 5_6 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 Page 129 27.301 75.690 11.932 46.968 58.606 57.223 37.237 72.956 14.630 87.592 8.591 28.054 19.468 53.067 40.563 4.088 81.483 20.259 42.654 23.473 23.695 84.804 30.677 53.427 77.769 12.701 79.178 53.807 74.608 1.124 39.439 1.909 76.815 51.372 48.878 35.421 8.595 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 Fig 5_6 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 Page 130 86.114 8.377 23.225 73.706 16.969 51.279 93.173 70.035 91.842 97.261 51.519 12.101 39.915 74.992 35.796 24.719 5.669 89.223 2.488 18.370 68.401 56.295 92.978 69.526 95.734 94.888 46.340 47.106 43.765 81.762 55.701 29.879 90.139 78.926 3.585 7.108 30.205 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 Fig 5_6 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 Page 131 96.758 77.143 22.047 94.019 28.662 34.148 33.934 3.654 69.944 58.653 9.323 59.167 61.141 27.693 27.569 17.436 20.672 97.094 13.170 15.559 43.435 60.276 59.324 25.197 15.977 89.203 15.336 94.903 92.788 22.444 25.108 89.546 99.587 47.154 83.565 28.249 81.302 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 Fig 5_6 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 Page 132 17.499 31.904 51.246 76.152 41.227 10.413 37.293 68.921 37.982 54.729 89.592 35.076 67.899 5.151 78.511 28.175 64.476 3.708 44.152 53.679 19.044 39.055 46.467 41.487 64.162 36.013 41.075 11.317 19.578 69.324 92.619 37.078 1.228 43.865 13.230 42.455 54.278 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 Fig 5_6 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 Page 133 50.523 11.376 92.260 5.252 0.969 27.337 73.152 6.120 5.848 1.327 70.596 9.556 45.479 24.275 28.600 84.533 70.742 70.087 48.696 6.755 11.162 60.012 26.333 80.486 52.631 63.411 81.714 96.496 76.641 24.169 50.775 27.165 35.546 43.035 32.417 36.514 70.372 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 Fig 5_6 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 Page 134 5.569 42.634 76.220 6.896 13.230 85.776 52.374 37.505 14.376 36.908 8.247 84.463 85.603 15.002 95.624 45.616 41.335 76.110 98.247 4.747 57.824 94.743 81.388 81.994 45.518 8.553 17.540 88.553 40.970 54.054 58.925 46.539 96.688 35.144 53.435 9.918 20.920 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 Fig 5_6 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 Page 135 5.809 47.423 35.296 42.717 55.670 19.759 28.320 70.672 15.383 73.936 12.007 91.493 72.183 16.754 49.318 66.926 98.142 31.312 12.443 6.695 48.851 0.996 47.665 2.905 59.920 94.204 99.593 95.065 47.639 9.512 15.985 53.449 56.935 51.281 96.166 12.605 71.040 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 Fig 5_6 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 Page 136 24.486 83.277 86.423 98.422 95.284 77.916 70.605 12.038 27.234 37.530 10.180 58.546 49.974 16.874 7.397 50.969 64.539 10.302 10.890 58.743 9.895 5.954 6.382 19.407 21.940 59.831 76.342 73.221 55.997 88.947 44.261 80.483 72.224 30.684 78.905 67.509 8.600 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 Fig 5_6 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 Page 137 49.510 79.547 35.834 87.040 89.727 94.380 37.014 6.601 1.777 87.983 71.140 12.079 98.873 29.883 21.974 4.827 36.265 41.381 26.767 96.096 17.723 99.988 52.093 6.670 44.249 32.576 78.895 74.933 11.481 46.403 83.533 60.991 25.950 19.367 48.031 15.677 13.747 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 Fig 5_6 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 Page 138 85.045 22.278 15.524 73.028 93.418 27.603 71.901 23.301 49.577 76.728 59.566 90.958 3.495 55.662 8.680 3.483 7.755 15.351 47.732 40.331 94.245 22.664 51.812 40.648 6.197 12.803 66.599 25.564 60.834 82.276 39.311 45.879 4.553 54.835 18.907 97.971 82.437 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 Fig 5_6 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 Page 139 90.807 21.272 32.014 67.535 80.838 22.972 71.030 36.500 31.652 74.513 44.255 47.003 22.245 84.586 41.248 44.909 36.398 81.896 51.106 49.200 48.495 76.670 10.034 30.770 15.980 55.913 35.324 70.815 74.819 33.294 53.252 65.627 54.566 85.266 33.162 35.404 8.238 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 Fig 5_6 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 Page 140 4.192 71.903 39.890 78.706 16.159 86.893 0.950 0.744 28.140 45.859 37.142 10.036 96.965 86.343 58.531 73.635 96.377 89.302 89.615 52.289 24.625 60.430 27.109 57.919 13.682 92.735 12.485 98.949 25.897 47.889 7.187 30.089 19.793 47.077 8.795 35.951 33.969 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 Fig 5_6 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 Page 141 9.745 36.696 62.110 55.605 73.838 72.146 52.570 60.180 30.677 26.205 56.557 19.979 15.820 8.847 44.604 76.250 35.955 2.523 89.932 28.690 15.009 88.881 54.587 62.898 96.067 84.677 82.691 43.144 93.471 18.642 77.113 3.217 55.337 39.223 58.822 29.175 11.369 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 Fig 5_6 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 Page 142 11.392 89.355 42.046 37.597 45.913 62.025 53.417 54.759 6.629 29.667 90.714 9.153 19.599 19.405 24.162 8.480 73.992 87.060 4.547 58.668 69.750 47.691 52.140 88.392 24.805 55.357 43.729 64.028 14.178 72.904 75.397 25.570 62.259 17.444 63.167 8.172 79.469 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 Fig 5_6 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 Page 143 16.583 62.931 86.098 46.250 53.646 95.251 65.849 73.050 19.412 74.329 47.042 6.472 78.876 5.711 76.222 26.568 57.850 64.614 51.373 13.207 8.342 15.401 27.386 81.246 90.798 52.956 43.506 8.241 16.122 51.677 87.710 32.706 14.609 73.808 78.956 68.254 69.059 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 Fig 5_6 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 Page 144 44.805 41.304 88.471 19.134 88.346 94.943 98.010 94.057 71.164 24.578 51.907 35.778 75.950 65.114 44.120 91.351 92.500 25.367 82.148 45.455 68.872 90.390 61.578 20.550 78.100 94.284 35.158 51.908 73.239 3.412 20.966 18.044 44.716 9.437 37.178 33.063 4.380 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 Fig 5_6 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 Page 145 35.188 27.120 75.544 59.766 79.027 11.322 35.716 44.141 55.442 27.067 36.641 80.809 9.215 82.097 49.681 99.605 43.674 70.230 77.704 37.958 5.388 29.612 11.197 8.801 50.578 29.241 53.517 60.015 66.419 86.580 64.395 1.608 13.700 39.939 61.373 92.726 51.260 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0 1 Fig 5_6 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 Page 146 97.089 36.867 6.702 24.156 73.509 87.511 33.371 55.605 37.192 32.976 99.279 7.422 10.681 37.237 12.811 40.293 48.434 21.611 90.871 77.676 75.129 50.886 44.095 61.709 15.281 45.703 75.408 55.219 7.076 68.135 6.480 4.165 5.002 13.182 28.321 78.511 0.693 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 1 Fig 5_6 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 Page 147 61.692 34.116 37.885 94.668 33.395 45.307 5.349 70.632 58.118 45.642 19.067 79.729 36.512 96.742 54.858 87.398 40.838 16.566 2.679 86.540 91.974 57.898 93.616 60.109 64.378 97.782 65.111 77.560 26.103 43.621 78.253 87.795 77.737 16.138 82.463 11.132 61.445 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 Fig 5_6 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 Page 148 87.812 81.764 19.563 33.454 0.831 99.292 69.966 97.573 54.149 57.365 38.411 70.715 60.044 24.951 62.690 17.942 18.568 22.799 82.320 16.349 87.910 59.880 42.452 31.531 38.134 30.247 9.268 54.272 12.710 20.400 15.717 0.522 2.165 35.280 33.976 2.996 34.571 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 Fig 5_6 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 Page 149 3.942 0.569 88.721 61.307 38.980 59.436 21.351 63.931 22.126 39.293 82.498 44.925 21.613 98.847 32.835 81.493 41.299 64.366 19.627 71.546 73.635 73.899 84.256 94.035 89.616 84.778 96.200 24.896 18.754 99.195 59.467 22.697 99.764 48.188 84.004 38.744 7.624 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 Fig 5_6 990 991 992 993 994 995 996 997 998 999 1000 Page 150 5.354 2.674 29.750 44.647 85.173 74.675 66.260 84.020 7.510 47.753 25.319 1 1 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 1 Fig 5_6 85 35 100 15 830 284 81.93 1000 170 100 to C 100 1000 to D 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 151 Fig 5_6 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 152 Fig 5_6 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 153 Fig 5_6 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 154 Fig 5_6 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 155 Fig 5_6 1 0 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 156 Fig 5_6 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 157 Fig 5_6 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 158 Fig 5_6 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 159 Fig 5_6 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 160 Fig 5_6 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 161 Fig 5_6 1 1 1 1 1 0 1 1 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 162 Fig 5_6 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 163 Fig 5_6 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 164 Fig 5_6 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 165 Fig 5_6 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 166 Fig 5_6 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 167 Fig 5_6 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 168 Fig 5_6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 169 Fig 5_6 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 170 Fig 5_6 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 171 Fig 5_6 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 172 Fig 5_6 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 173 Fig 5_6 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 174 Fig 5_6 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 175 Fig 5_6 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 176 Fig 5_6 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 177 Fig 5_6 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Page 178 Fig 5_7 52 Cards in Four Suits spade diamond club heart 1 Ace Ace Ace Ace Ace 2 King King King King King 3 Queen Queen Queen Queen Queen 4 Jack Jack Jack Jack Jack 5 Ten Ten Ten Ten Ten 6 Nine Nine Nine Nine Nine 7 Eight Eight Eight Eight Eight 8 Seven Seven Seven Seven Seven 9 Six Six Six Six Six 10 Five Five Five Five Five 11 Four Four Four Four Four 12 Three Three Three Three Three 13 Two Two Two Two Two An ordinary poker deck of cards has 52 cards in four suits: spades, diamonds, clubs, hearts. Each suit has 13 cards: two to ten, three face cards (J, Q, K) and an ace. With complete shuffling, each card is equally likely to be drawn from the top of the deck. Enter any number to generate new random draw 3 Value Suit Copy and paste special value Jack Club 1 2 Enter any number in the yellow cell to 3 generate a random draw from the card 4 deck and touch return. The result appears 5 immediately. 6 7 Record 25 draws (by copying and pasting 8 values in columns) and compare the 9 pattern to that predicted by probability 10 theory. 11 12 The simulation is at cell AA1. Page 179 values in columns) and compare the pattern to that predicted by probability theory. Fig 5_7 The simulation is at cell AA1. Page 180 13 14 15 16 17 18 19 20 21 22 23 24 25 An ordinary poker deck of cards has 52 cards in four suits: spades, diamonds, clubs, hearts. Each suit has 13 cards: two to ten, three face cards (J, Q, K) and an ace. With complete shuffling, each card is equally likely to be drawn from the top of the deck. Fig 5_7 Probability of Ace? Heart Ace of Hearts? Ace or Hearts? opy and paste special values down these columns. Value Suit Page 181 Fig 5_7 count suits rel freq probability heart 0.25 diamond 0.25 spade 0.25 club 0.25 Page 182 Fig 5_7 How does a We create a columns, we example, "Ac In a separate =ROUND((RA between one Use the VLOO from the 52 r random num required valu 1) the cel 2) the ran 3) the col Page 183 Fig 5_7 Page 184 Fig 5_7 36 How does a spreadsheet draw a card from a deck? We create a column of numbers 1 to 52. In the next two columns, we enter the suit and value of the card, for example, "Ace" and "Heart". See Column AC. In a separate cell, we enter =ROUND((RAND()*52)+0.5,0) to pick a random number between one and 52. Use the VLOOKUP function to pull the specific card label from the 52 rows of the table of values using the random number to pick the row. VLOOKUP has three required values: 1) the cell with the random number 2) the range of cells with the table of card labels 3) the column to pick the label. Page 185 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Fig 5_7 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Page 186 Fig 5_7 2 Spade 3 4 5 6 7 8 9 10 Jack Queen King Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Ace 2 3 4 5 6 7 8 9 10 Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Club Club Club Club Club Club Club Club Club Page 187 Fig 5_7 Jack Queen King Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Ace Club Club Club Club Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Page 188 Fig 5_8 Take Two Enter any number to generate two random draws without replacement. 2 Value Suit King Heart The probabilities associated Ace Spade with the second card drawn depend on what card was drawn first. The simulation is at ce The probability of a heart given a succession of hearts. If all prior draws were a heart, the probability of getting a heart on the next draw declines. Draw # 1 13/52 0.250 2 12/51 0.235 3 11/50 0.220 4 10/49 0.204 5 9/48 0.188 Probability of heart draw being a heart, given that all prior draws were hearts. Page 189 6 8/47 0.170 Fig 5_8 Page 190 Fig 5_8 Page 191 Fig 5_8 eplacement. obabilities associated e second card drawn d on what card was first. The spreadsheet ha one after the other. A random number f first card. he simulation is at cell Q5. A second random n picks randomly from first pick. In this way, the first selected in the seco other cards are equ Page 192 Fig 5_8 Page 193 Fig 5_8 Page 194 Fig 5_8 positions 1 position 2 first draw 51 3 second draw 75 4 5 6 7 The spreadsheet has two lists of the cards, 8 one after the other. 9 10 A random number from one to 52 picks the 11 first card. 12 13 A second random number of one to 51 14 picks randomly from the 51 cards below the 15 first pick. 16 17 In this way, the first card drawn cannot be 18 selected in the second draw but all of the 19 other cards are equally likely to be drawn. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Page 195 value 2 3 4 5 6 7 8 9 10 Jack Queen King Ace 2 3 4 5 6 7 8 9 10 Jack Queen King Ace 2 3 4 5 6 7 8 9 10 suit Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Diamond Club Club Club Club Club Club Club Club Club Fig 5_8 36 Jack 37 Queen 38 King 39 Ace 40 2 41 3 42 4 43 5 44 6 45 7 46 8 47 9 48 10 49 Jack 50 Queen 51 King 52 Ace 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 Jack 11 Queen 12 King 13 Ace 14 2 15 3 16 4 17 5 18 6 19 7 20 8 Page 196 Club Club Club Club Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Spade Diamond Diamond Diamond Diamond Diamond Diamond Diamond Fig 5_8 21 9 22 10 23 Jack 24 Queen 25 King 26 Ace 27 2 28 3 29 4 30 5 31 6 32 7 33 8 34 9 35 10 36 Jack 37 Queen 38 King 39 Ace 40 2 41 3 42 4 43 5 44 6 45 7 46 8 47 9 48 10 49 Jack 50 Queen 51 King 52 Ace Page 197 Diamond Diamond Diamond Diamond Diamond Diamond Club Club Club Club Club Club Club Club Club Club Club Club Club Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Heart Fig 5_9 Horse Race 1997 Kentucky Derby Odds from Daily Racing Form, 1997 1 1/4 mile race for three year olds carrying 126 lb. pole fin Horse Odds Probability 5 1 Silver Charm 4.00 0.1612 4 2 Captain Bodgit 3.10 0.1966 13 3 Free House 10.60 0.0695 7 4 Pulpit 5.70 0.1203 1 5 Crypto Star 4.80 0.1390 2 6 Phantom on Tour 19.50 0.0393 9 7 Jack Flash 20.90 0.0368 8 8 Hello 9.60 0.0760 3 9 Concerto 10.80 0.0683 6 10 Celtic Warrior 37.10 0.0212 12 11 Crimson Classic 80.10 0.0099 10 12 Shammy Davis 20.90 0.0368 11 13 Deeds Not Words 32.40 0.0241 total 0.9990 t = 0.19 Two to one losses. The take out rate is the proportion of bets th Odds: dollars paid on $1 bet (There is a $2 minimum bet.) That is, the winner gets the value of the bet plus the odds. One dollar gets $4 gross on three to one odds. The mutual pool for the '97 Derby was $23,592,756. The winning horse earned $700,000, 2nd got $170,000, 3rd got $85,000, and 4th got $45,000: Total purse $1M. A $2 bet on Silver Charm to win paid $10. A $2 bet on Silver Charm or Captain Bodgit to place (1st or 2nd) pa A $2 bet on Silver Charm to show (1st, 2nd, or 3rd) paid $4.20. A $2 show bet on Captain Bodgit paid $3.80. A $2 show bet on Free House paid $5.80. Place and Show bets are put in separate pools. Each pool pays ac Comparison of Betting Probabilities with the Frequency of W Page 198 Fig 5_9 reference: Peter Asch and Richard E. Quandt, Racetrack Betting: the Professors' Guide to Strategies, (Auburn House, Dover, MA, 198 Asch-Quandt, Race Track Betting, p. 112, Table 6-1 Group Favorites 2nd low odds 3rd low odds 4th low odds 5th low odds 6th low odds 7th low odds 8th low odds 9th low adds # of horses 729 729 729 724 692 598 431 289 165 Objective p 0.361 0.218 0.170 0.115 0.071 0.050 0.030 0.017 0.006 from Asch, Malkiel, Quandt, "Racetrack Betting and Informed Behavior," Journal of Financial Economics 10 (1982) pp. 1 Subjective p are the probabilities defined by the proportion bet on each horse, averaged over all the races. Objective probabilities are the observed proportion of the horses w The study is of 5,805 horses in 729 races in Atlantic City in1978. Studies at other tracks give comparable results. With some consistency, bettors underbet the favorites. A Fictitious Horse Race The table below describes a fictitious race of seven horses. The column reports the total amount bet. The second column defines probability of winning implied by the pattern of the bets. The thi defines the odds ($ of net payout per dollar bet). Try changing th amounts bet to see the effect on the probabilities and the odds. change continuously to race time as the pattern of betting shifts. bets are paid according to the final odds, not the odds posted wh ticket is bought. Page 199 change continuously to race time as the pattern of betting shifts. bets are paid according to the final odds, not the odds posted wh ticket is bought. Fig 5_9 Suppose total dollars bet on seven horses ar This pattern of bets implies a set of probab Enter new values of the bets to see how the horse $s bet Patty Cake $150 Jodie's Hope $450 Silver Blur $950 Bob Along $600 Outstanding in his Field $750 Monday Monday $1,100 Goodbye $500 sum $4,500 Change the rate of "take" to see the effect o Take for track 20% implies $s to track $900 $s to bettors $3,600 Page 200 Fig 5_9 Two to one odds means one win for each two losses. Treat the dollars bet on each horse to win its race as probabilities. The odds are then: D = ( 1 - p(x) )/ p(x) Given the odds D, probablity is p(x) = 1/(1+ D) proportion of bets that goes to the track. inimum bet.) plus the odds. $170,000, purse $1M. ($4 odds on each $1 plus the $2 bet.) place (1st or 2nd) paid $4.80. r 3rd) paid $4.20. ls. Each pool pays according to the amount bet in that pool. the Frequency of Wins Page 201 Fig 5_9 Racetrack Betting: House, Dover, MA, 1986). Subjective p 0.325 0.205 0.145 0.104 0.072 0.048 0.034 0.025 0.018 omics 10 (1982) pp. 187-94. he proportion ortion of the horses who won. tlantic City in1978. favorites. f seven horses. The first econd column defines the n of the bets. The third column bet). Try changing the bilities and the odds. Odds tern of betting shifts. Winning t the odds posted when a Page 202 tern of betting shifts. Winning t the odds posted when a Fig 5_9 et on seven horses are as follows: mplies a set of probabilities and odds. e bets to see how the probabilities and odds change. # prob odds: D to 1 implied p 1 0.0333 23.0 0.0333 2 0.1000 7.0 0.1000 3 0.2111 2.7 0.2162 4 0.1333 5.0 0.1333 5 0.1667 3.8 0.1667 6 0.2444 2.2 0.2500 7 0.1111 6.2 0.1111 1.0000 1.0107 e" to see the effect on the odds. Odds =( ((1-t)*B)/Bi) - 1 p(x) = (1-t)/(odds +1) t is take out rate B is total bet in pool Bi is total bet on horse i Page 203 Define a universe of equally likely outcomes. Count all the events in the universe. Count the events in the outcome of interest. The ratio is the mathematical probability. Use a tree or table to help define the elements of the universe. Use complements when convenient. Watch for changing conditions as when cards are drawn without replacement. Probability = 1/ ( 1 + odds ) Addition and multiplication are faster ways of counting. There are three conceptions of probability. 1. Mathematical probability: counting equally like 2. Subjective probability: probability that you wil 3. Probability as frequency: the proportion of out s. t. ents of the ds are drawn of counting. bability. ting equally likely events. lity that you will get an MBA. roportion of outcomes observed ex post. ...
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This note was uploaded on 07/12/2008 for the course ECON 150 taught by Professor Renhoff during the Spring '08 term at Vanderbilt.

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