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}}}{EXCHG {PARA 206 "" 0 "" {TEXT 277 0 "" }}{PARA 209 "" 0 "" {TEXT 207 0 "" }{TEXT 207 0 "" }{TEXT 207 0 "" }{TEXT 207 0 "" }{TEXT 207 42 "Lab 08 - Counting and Probability in Maple" }{TEXT 207 0 "" }{TEXT 207 0 "" }}{PARA 206 "" 0 "" {TEXT 277 29 "You may work with ONE part ner" }}{PARA 206 "" 0 "" {TEXT 212 5 "due: " }{TEXT 277 31 "Tuesday, N ovember 03 at 9:00 am" }}{PARA 206 "" 0 "" {TEXT 277 9 "15 points" }} {PARA 206 "" 0 "" {TEXT 213 0 "" }{TEXT 277 0 "" }{TEXT 214 0 "" } {TEXT 215 31 "Upload your completed worksheet" }{TEXT 216 0 "" }{TEXT 277 1 " " }{TEXT 217 0 "" }{TEXT 218 24 "in Moodle with filename:" }} {PARA 206 "" 0 "" {TEXT 277 0 "" }{TEXT 219 10 "ma116Lab08" }{TEXT 220 9 "your-name" }{TEXT 277 0 "" }{TEXT 221 4 ".mws" }{TEXT 277 5 " ( or " }{TEXT 222 3 ".mw" }{TEXT 277 1 ")" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 223 0 "" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 62 "1. Be sure you have opened this Maple workshe et as a Maple 13" }{TEXT 276 1 " " }{TEXT 225 18 "Classic Worksheet." }}{PARA 0 "" 0 "" {TEXT 276 77 " (a) You should save this file to y our U: drive after clicking on the link" }}{PARA 0 "" 0 "" {TEXT 276 42 " (b) Start \"Maple 13 Classic Worksheet\"" }}{PARA 0 "" 0 "" {TEXT 276 57 " (c) Open this worksheet, on your U: drive, from Mapl e" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 3 " " }{TEXT 226 30 "(d) Maple is like the movie, " }{TEXT 227 14 "50 Fir st Dates" }{TEXT 228 239 ". When you exit a Maple worksheet, and then re-open it, the memory is emptied, so even though you can still see a previous result in blue, Maple doesn't have this in memory unless you re-execute the execution group or the entire worksheet" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 207 103 "2. Once you have opened this worksheet, do a \"Save\" to your U: drive, and then do them every so often." }} {PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 211 "" 0 "" {TEXT 276 72 "Here, we just want to get familiar with the factorial function in \+ Maple." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 229 3 "(a)" }{TEXT 276 44 " Have Maple calculat e 47! (\"47 Factorial\")." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 157 "Remember that P(n,r) = n!/(n-r)!. Notice below how Maple cal culates P(47,4) as a quotient of factorials, and where the denominator depends on a subtraction." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "47!/(47-4)!;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 230 3 "(b)" }{TEXT 276 111 " Ha ve Maple calculate P(26,3) as a quotient of factorials, and where the \+ denominator depends on a subtraction." }}{PARA 0 "" 0 "" {TEXT 276 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 276 0 "" }{TEXT 207 0 "" }{TEXT 207 24 "4. The Birthday P roblem" }{TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 231 0 "" }{TEXT 276 105 "Note that we will make some simpli fying assumptions, to make our task a bit easier. The biggest will be " }{TEXT 232 71 "that all years have 365 days and that all birthdays \+ are equally likely." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 233 5 "(a) " }{TEXT 276 314 "First, g iven a group of 4 people, use Maple to determine the total number of w ays that birthdays could be associated with each person. For example, person A and person C could have both been born on 01/11, while B's b irthday could be 7/20 and D's birthday could be 04/08. Assign your ca lculation to the variable " }{TEXT 234 5 "total" }{TEXT 276 1 "." }} {PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "totalForFourPeople := " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 235 5 "(b) " }{TEXT 276 109 "Next, use Maple to determine how many ways birthdays could be associated with persons A, B, C, and D so that " }{TEXT 236 2 "no" }{TEXT 276 84 " two people would share the same birthday? Assign your calculation to the variab le " }{TEXT 237 5 "noTwo" }{TEXT 276 57 ". Use the factorial function in the same way you did in " }{TEXT 238 2 "#3" }{TEXT 276 19 " above, as a ratio." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "noTwo := " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 239 5 "(c) " }{TEXT 276 84 "So, how many w ays birthdays could be associated with persons A, B, C, and D so that " }{TEXT 240 8 "at least" }{TEXT 276 80 " two people would share the s ame birthday? Use the variables you defined above." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "atLeastTw o := " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 241 5 "(d) " }{TEXT 276 94 "What is the probability that at lea st two people out of A, B, C, and D have the same birthday?" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ev alf( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 242 4 "(e) " }{TEXT 276 42 " In that problem above, we only disc ussed " }{TEXT 243 0 "" }{TEXT 244 0 "" }{TEXT 245 4 "four" }{TEXT 246 0 "" }{TEXT 276 75 " people. Now, we would like to generalize the problem, and we'll consider " }{TEXT 247 0 "" }{TEXT 248 0 "" }{TEXT 249 1 "n" }{TEXT 250 0 "" }{TEXT 251 0 "" }{TEXT 276 55 " people. In p articular, we want to determine how large " }{TEXT 252 1 "n" }{TEXT 276 50 " must be, so that in any randomly chosen group of " }{TEXT 253 1 "n" }{TEXT 276 78 " people, the probability that 2 or more have \+ the same birthday is 50% or more." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }} {PARA 0 "" 0 "" {TEXT 276 287 "Consider the calculations you performed above for n=4 people, and consider how you will generalize them and p ut them in a loop over n. Then look for the first value of n that giv es a percentage over 50%. (Note that the loop takes n from 4 up to 30, and the value of n is in this range)." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "for n from 4 to 30\n" }{MPLTEXT 1 0 4 "do;\n" }{MPLTEXT 1 0 23 " totalForNPeople := \n" } {MPLTEXT 1 0 14 " noTwo := \n" }{MPLTEXT 1 0 19 " atLeastTwo := \+ \n" }{MPLTEXT 1 0 14 " n; evalf( \n" }{MPLTEXT 1 0 7 "end do;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 213 "" 0 "" {TEXT 276 0 "" }{TEXT 207 0 "" }{TEXT 207 7 "Answer:" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 254 0 "" } {TEXT 255 0 "" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 256 0 "" }}{PARA 214 "" 0 "" {TEXT 207 26 "5 . Simulating coin tosses" }{TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 257 0 "" }{TEXT 276 170 "Next, we wil l use Maple to study coin tossing. In particular, we want to compare \+ the theoretical, expected results with the actual simulation of tossin g a coin 10 times." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 30 "Let's begin by determing some " }{TEXT 258 11 "theoreti cal" }{TEXT 276 8 " values." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "(a) " }{TEXT 276 74 " What is the total number \+ of possible outcomes of tossing a coin 10 times?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "total := " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "(b) " }{TEXT 276 60 " In how m any of these outcomes are exactly 5 heads obtained?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fiveHeads := " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "(c) " }{TEXT 276 129 " U sing your results in (a) and (b), what is the probability that exactly 5 heads will be obtained when a coin is tossed 10 times?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "evalf( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 116 "Now, let's simulate co in tossing and compare our experimental results to the theoretical res ult we determined above." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 " " 0 "" {TEXT 276 59 "First, use the Maple help and read up on the rand function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?rand" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 65 " So, it seems that, you can get a random 12 digit number by using " } {TEXT 262 7 "rand( )" }{TEXT 276 120 ". To restrict the range of the \+ output, you can use a parameter, and to do this repeatedly you can ass ign the output of " }{TEXT 263 8 "rand ( )" }{TEXT 276 45 " to a varia ble, where below we used the name " }{TEXT 264 4 "flip" }{TEXT 276 1 " ." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "(d) " }{TEXT 276 61 " If I try it to simulate a coin flip, I would define flip as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "flip := " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 134 "And then I would just call flip( ) to simulate a coin toss. I could \+ choose 0 to correspond to \"tails\" and 1 to correspond to \"heads.\"" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 276 21 "Let' s try one \"flip.\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "flip( );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "(e) " }{TEXT 276 60 " Write a loop which would simulate flippin g a coin 10 times." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "for i from 1 to 10\n" }{MPLTEXT 1 0 4 "do;\n" }{MPLTEXT 1 0 16 " _________ __; \n" }{MPLTEXT 1 0 7 "end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 267 5 "(f) " }{TEXT 276 128 "Modify your loop so that rather than outputing each flip, it suppresses output an d just determines how many heads are obtained.\n" }{TEXT 276 61 "You'l l need a variable to keep a running total... and use a " }{TEXT 268 1 ":" }{TEXT 276 16 " , instead of a " }{TEXT 269 1 ";" }{TEXT 276 20 " to suppress output." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nu mH := ____ ;\n" }{MPLTEXT 1 0 19 "for i from 1 to 10\n" }{MPLTEXT 1 0 4 "do;\n" }{MPLTEXT 1 0 30 " ______ := ______ + ______ ;\n" }{MPLTEXT 1 0 8 "end do:\n" }{MPLTEXT 1 0 5 "numH;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 270 5 "(g) " }{TEXT 276 170 "S o, your loop in (f) simulates one round of tossing a coin 10 times. P ut an outer loop around it, which runs 1000 times, and count how many \+ times 5 heads are obtained. " }{TEXT 271 12 "START SMALL." }{TEXT 276 114 " Test your code on small sized loops first. Don't forget to suppress output (see the note above about : vs. ;).." }}{PARA 0 "" 0 "" {TEXT 276 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fiveH : = _______ ;\n" }{MPLTEXT 1 0 24 "for j from 1 to _______\n" }{MPLTEXT 1 0 4 "do;\n" }{MPLTEXT 1 0 19 " numH := _______:\n" }{MPLTEXT 1 0 21 " for i from 1 to 10\n" }{MPLTEXT 1 0 6 " do;\n" }{MPLTEXT 1 0 35 " _______ := _______ + _______ ;\n" }{MPLTEXT 1 0 10 " end do: \n" }{MPLTEXT 1 0 58 " if (numH=5) then _______ := _______ + _______ \+ ; end if;\n" }{MPLTEXT 1 0 8 "end do:\n" }{MPLTEXT 1 0 5 "tenH;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 0 "" 0 "" {TEXT 272 5 "(h) " }{TEXT 276 140 "S o, in your simulation, how many times did you get 5 heads? How close \+ is that percentage to the theoretical value you derived in (c) above?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 0 "" }}{PARA 215 "" 0 "" {TEXT 276 0 "" }{TEXT 207 0 "" }{TEXT 207 7 "Answer:" }}{PARA 0 "" 0 "" {TEXT 276 0 "" }{TEXT 273 0 "" }{TEXT 274 0 "" }{TEXT 275 0 "" }} {PARA 0 "" 0 "" {TEXT 276 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }