\(P(red \ OR \ blue) = P(red) + P(blue)\)
Total Marbles = 138
\(P(red) = \frac{54}{138} = \frac{9}{23}\) = .03913
\(P(blue) = \frac{75}{138}\) = 0.5435
\(P(red)+P(blue)\) = 0.9348
\(P(red) = \frac{P(red)}{P(total)} = \frac{20}{80} =\) 0.25
\(P(not \ male \cup not \ in \ parent's \ house)= 1 - P(male \cap in \ parent's \ house)\)
\(P(male) = \frac{671}{1399} = 0.4796\)
\(P(in \ parent's \ house) = \frac{467}{1399} = 0.3338\)
\(0.4796 \times 0.3338 = 0.16\)
\(1 - 0.16 =\) 0.84
Dependent
\[\left (\begin{array}{c}
8 \\
3 \end{array} \right) \times
\left (\begin{array}{c}
7 \\
3 \end{array} \right) \times 3
\]
\[\left (\begin{array}{c}
8 \\
3 \end{array} \right) = \frac{8!}{3!(8-3)!} = 56
\] \[\left (\begin{array}{c}
7 \\
3 \end{array} \right) = \frac{7!}{3!(7-3)!} = 35
\] \(56 \times 35 \times 3 =\) 5880
56*35*3## [1] 5880Independent
\[\frac{14!}{(14-8)!} = 121080960\]
Total number of ways 4 jelly beans can be drawn from 22. \[\left (\begin{array}{c} 22 \\ 4 \end{array} \right)\]
Number of ways to pick 1 orange jelly beans \[\left (\begin{array}{c} 4 \\ 1 \end{array} \right)\]
Number of ways to pick 3 green jelly beans \[\left (\begin{array}{c} 9 \\ 3 \end{array} \right)\]
Probability of choosing 3 green and 1 orange jelly beans \[\frac{\left (\begin{array}{c} 4 \\ 1 \end{array} \right) \left (\begin{array}{c} 9 \\ 3 \end{array} \right)}{\left (\begin{array}{c} 22 \\ 4 \end{array} \right)} = 0.0459 \]
factorial(11)/factorial(7)## [1] 792033% of subscribers to a fitness magazine are under the age of 34
Step 1)
Total outcomes = \(2^4 = 16\)
Number of ways to get 3 heads = \[\left (\begin{array}{c} 4 \\ 3 \end{array} \right) = 4\]
P(3 heads) = \(\frac{4}{16} = \frac{1}{4} = (+\$97)\) P(not 3 heads) = \(\frac{3}{4} = (-\$30)\)
\((0.25 \times 97) + (.75 \times -30) = \$1.75\)
Step 2)
\(\$1.75 \times 559 = \$978.25\)
outcome <- c()
coin <- c(0,1) #tails = 0, heads = 1
prob <- 0.5 #fair coin
n <- 559 #number of times to flip
i <- 1
while(i <= n){
set <- sample(coin, size = 4, replace = TRUE, prob = c(.5,.5))
if(sum(set) == 3) {outcome <- c(outcome, sum(outcome[i-1], 97))}
else {outcome <- c(outcome, sum(outcome[i-1], -30))}
i <- i + 1
}
outcome## [1] -30 -60 37 7 -23 74 44 14 111 81 178 148 118 88
## [15] 58 155 125 95 192 162 132 102 199 169 139 236 206 176
## [29] 146 116 86 183 153 123 93 63 33 3 -27 -57 -87 -117
## [43] -147 -177 -207 -110 -13 -43 -73 24 -6 -36 -66 -96 -126 -156
## [57] -59 -89 -119 -149 -179 -209 -112 -142 -172 -75 -105 -135 -38 59
## [71] 29 -1 -31 -61 -91 -121 -24 -54 43 13 -17 80 50 147
## [85] 117 87 57 27 124 94 64 34 131 101 71 41 11 -19
## [99] -49 -79 -109 -139 -42 55 152 122 92 62 32 2 -28 -58
## [113] 39 136 106 76 46 143 113 83 53 23 -7 90 60 30
## [127] 127 97 67 37 7 -23 -53 -83 14 -16 -46 -76 -106 -9
## [141] -39 -69 -99 -129 -32 -62 -92 -122 -25 -55 -85 12 -18 -48
## [155] -78 -108 -138 -168 -198 -228 -258 -288 -318 -221 -251 -154 -184 -214
## [169] -244 -274 -304 -207 -237 -267 -297 -200 -230 -260 -290 -320 -350 -253
## [183] -156 -59 -89 -119 -149 -179 -82 -112 -142 -172 -202 -232 -135 -38
## [197] 59 29 126 223 320 417 514 484 454 424 394 491 461 558
## [211] 528 498 468 438 408 505 475 445 415 385 355 325 295 265
## [225] 362 332 302 272 369 339 309 406 376 346 316 286 256 226
## [239] 196 166 263 233 203 173 270 367 464 434 404 374 471 568
## [253] 538 508 478 448 418 388 358 328 425 522 619 589 559 656
## [267] 753 850 947 1044 1014 1111 1081 1051 1021 991 961 931 901 871
## [281] 841 811 781 751 721 691 661 758 728 698 668 638 608 578
## [295] 675 645 742 712 809 779 749 719 689 659 629 599 696 793
## [309] 763 860 957 1054 1024 994 1091 1061 1031 1001 971 941 911 1008
## [323] 1105 1202 1172 1142 1239 1209 1179 1149 1119 1089 1059 1029 999 1096
## [337] 1066 1163 1133 1230 1327 1297 1267 1237 1207 1304 1274 1244 1214 1184
## [351] 1154 1124 1094 1064 1034 1004 1101 1071 1168 1138 1108 1078 1048 1018
## [365] 1115 1085 1055 1025 995 965 935 905 875 845 815 912 882 852
## [379] 949 1046 1143 1240 1210 1180 1150 1120 1090 1060 1030 1000 970 940
## [393] 1037 1007 1104 1074 1044 1014 984 954 924 1021 991 961 931 1028
## [407] 1125 1222 1192 1162 1132 1102 1072 1169 1139 1236 1206 1303 1400 1497
## [421] 1594 1564 1534 1504 1474 1444 1414 1384 1354 1324 1294 1264 1234 1331
## [435] 1301 1271 1241 1211 1181 1151 1121 1091 1061 1158 1128 1225 1195 1165
## [449] 1262 1359 1329 1299 1396 1493 1463 1433 1403 1500 1597 1694 1791 1888
## [463] 1985 1955 1925 1895 1865 1835 1805 1775 1745 1715 1685 1655 1625 1722
## [477] 1819 1789 1886 1856 1826 1796 1766 1736 1706 1676 1646 1616 1586 1556
## [491] 1526 1623 1593 1690 1660 1630 1727 1697 1667 1637 1607 1704 1674 1644
## [505] 1741 1711 1681 1651 1621 1591 1561 1531 1501 1471 1568 1665 1635 1605
## [519] 1702 1672 1769 1739 1709 1806 1903 1873 1970 1940 1910 2007 2104 2074
## [533] 2044 2014 1984 1954 1924 1894 1991 1961 2058 2028 1998 1968 1938 1908
## [547] 1878 1848 1818 1788 1885 1982 1952 2049 2146 2116 2086 2183 2153Step 1)
Total outcomes = \(2^9 = 512\)
Number of ways to get 4 tails or less = \[\left (\begin{array}{c} 9 \\ 4 \end{array} \right) + \left (\begin{array}{c} 9 \\ 3 \end{array} \right) + \left (\begin{array}{c} 9 \\ 2 \end{array} \right) + \left (\begin{array}{c} 9 \\ 1 \end{array} \right) + 1 = 256 \]
P(4 tails or less) = \(\frac{256}{512} = \frac{1}{2} = (+\$23)\) P(not 4 tails or less) = \(\frac{1}{2} = (-\$26)\)
\((0.5 \times 23) + (.5 \times -26) = \$1.50\)
Step 2)
\(\$1.75 \times 994 = \$-1,491\)
outcome <- c()
coin <- c(0,1) #tails = 0, heads = 1
prob <- 0.5 #fair coin
n <- 994 #number of times to flip
i <- 1
while(i <= n){
set <- sample(coin, size = 9, replace = TRUE, prob = c(.5,.5))
if(sum(set) >= 5) {outcome <- c(outcome, sum(outcome[i-1], 23))}
else {outcome <- c(outcome, sum(outcome[i-1], -26))}
i <- i + 1
}
outcome## [1] 23 -3 -29 -55 -81 -58 -35 -12 -38 -15 8
## [12] 31 5 -21 2 -24 -50 -76 -53 -79 -105 -82
## [23] -59 -85 -111 -88 -114 -140 -117 -143 -120 -97 -74
## [34] -51 -77 -103 -80 -57 -83 -60 -37 -63 -40 -17
## [45] 6 29 52 75 49 72 46 69 43 17 -9
## [56] -35 -61 -38 -15 -41 -18 -44 -21 2 -24 -1
## [67] -27 -53 -30 -56 -82 -108 -134 -111 -88 -114 -91
## [78] -117 -143 -169 -146 -123 -100 -126 -103 -80 -106 -83
## [89] -109 -86 -63 -89 -66 -43 -20 -46 -72 -98 -124
## [100] -101 -127 -153 -130 -107 -84 -61 -38 -15 8 -18
## [111] -44 -70 -96 -122 -99 -76 -102 -79 -105 -82 -108
## [122] -134 -160 -137 -114 -91 -117 -143 -169 -146 -123 -149
## [133] -126 -103 -80 -57 -34 -11 -37 -14 9 -17 -43
## [144] -20 -46 -23 -49 -26 -3 -29 -6 -32 -9 14
## [155] -12 11 34 57 31 5 -21 2 -24 -50 -27
## [166] -4 -30 -56 -33 -10 -36 -62 -88 -114 -140 -166
## [177] -192 -218 -244 -270 -296 -273 -299 -325 -351 -377 -403
## [188] -380 -406 -383 -409 -386 -412 -389 -366 -343 -369 -346
## [199] -372 -349 -375 -352 -329 -355 -381 -407 -384 -410 -387
## [210] -364 -390 -367 -393 -419 -396 -422 -399 -376 -402 -428
## [221] -454 -431 -408 -385 -362 -339 -365 -342 -319 -345 -322
## [232] -348 -374 -400 -426 -452 -429 -455 -481 -458 -484 -461
## [243] -438 -415 -441 -418 -395 -421 -398 -424 -401 -378 -355
## [254] -381 -407 -433 -459 -485 -511 -488 -514 -540 -517 -543
## [265] -569 -595 -572 -598 -624 -650 -676 -653 -630 -607 -633
## [276] -659 -636 -613 -639 -665 -642 -619 -596 -622 -648 -625
## [287] -602 -579 -556 -533 -510 -487 -464 -441 -418 -395 -372
## [298] -349 -375 -401 -378 -404 -430 -407 -433 -410 -436 -413
## [309] -390 -416 -442 -419 -445 -422 -448 -425 -451 -428 -454
## [320] -480 -506 -483 -509 -535 -561 -538 -564 -590 -616 -642
## [331] -619 -596 -573 -550 -576 -553 -530 -556 -533 -559 -536
## [342] -513 -539 -516 -493 -470 -447 -424 -450 -427 -404 -430
## [353] -407 -433 -459 -485 -511 -488 -465 -491 -517 -494 -471
## [364] -497 -523 -549 -575 -552 -578 -555 -532 -509 -535 -512
## [375] -538 -564 -590 -616 -593 -619 -596 -573 -550 -576 -602
## [386] -579 -556 -582 -608 -634 -660 -637 -663 -689 -666 -692
## [397] -669 -646 -672 -649 -626 -652 -678 -655 -632 -609 -586
## [408] -612 -638 -615 -592 -569 -595 -572 -549 -526 -552 -578
## [419] -555 -532 -509 -486 -463 -440 -417 -443 -420 -397 -423
## [430] -400 -377 -354 -380 -357 -334 -311 -337 -363 -389 -366
## [441] -343 -320 -346 -372 -349 -375 -401 -378 -404 -430 -456
## [452] -433 -410 -436 -413 -439 -465 -491 -517 -543 -569 -546
## [463] -572 -549 -526 -552 -529 -506 -532 -509 -486 -512 -489
## [474] -466 -443 -420 -446 -472 -449 -426 -452 -478 -455 -481
## [485] -507 -533 -559 -536 -562 -588 -565 -542 -568 -545 -571
## [496] -597 -623 -600 -626 -603 -629 -655 -632 -658 -684 -710
## [507] -687 -664 -641 -618 -644 -621 -598 -624 -650 -627 -653
## [518] -679 -705 -682 -708 -685 -662 -639 -665 -691 -717 -694
## [529] -720 -697 -674 -700 -726 -703 -729 -755 -732 -758 -784
## [540] -810 -836 -862 -888 -865 -842 -868 -845 -822 -799 -776
## [551] -753 -730 -756 -733 -710 -736 -713 -739 -765 -791 -768
## [562] -794 -820 -846 -872 -898 -875 -852 -829 -855 -881 -907
## [573] -884 -910 -936 -913 -890 -916 -893 -919 -945 -971 -948
## [584] -974 -951 -977 -954 -931 -957 -934 -911 -888 -865 -842
## [595] -868 -845 -822 -848 -874 -900 -926 -903 -929 -955 -932
## [606] -909 -935 -961 -938 -964 -990 -967 -993 -1019 -996 -1022
## [617] -1048 -1074 -1100 -1126 -1152 -1129 -1155 -1181 -1207 -1184 -1210
## [628] -1187 -1164 -1141 -1167 -1144 -1121 -1147 -1173 -1150 -1127 -1153
## [639] -1130 -1156 -1182 -1159 -1185 -1211 -1188 -1165 -1142 -1119 -1096
## [650] -1073 -1050 -1076 -1102 -1128 -1154 -1180 -1157 -1134 -1111 -1137
## [661] -1114 -1140 -1166 -1143 -1169 -1195 -1172 -1198 -1224 -1250 -1276
## [672] -1302 -1328 -1305 -1282 -1308 -1334 -1311 -1288 -1314 -1291 -1268
## [683] -1245 -1222 -1199 -1225 -1202 -1228 -1205 -1182 -1159 -1136 -1113
## [694] -1139 -1165 -1191 -1217 -1194 -1220 -1197 -1174 -1151 -1177 -1203
## [705] -1180 -1157 -1183 -1209 -1235 -1261 -1287 -1264 -1290 -1267 -1293
## [716] -1270 -1247 -1224 -1201 -1178 -1155 -1132 -1109 -1086 -1112 -1138
## [727] -1164 -1190 -1216 -1242 -1268 -1294 -1320 -1297 -1274 -1300 -1326
## [738] -1303 -1329 -1306 -1283 -1260 -1286 -1263 -1289 -1266 -1292 -1269
## [749] -1246 -1272 -1249 -1226 -1252 -1229 -1206 -1183 -1160 -1186 -1163
## [760] -1189 -1166 -1143 -1120 -1097 -1123 -1149 -1175 -1152 -1129 -1106
## [771] -1083 -1109 -1135 -1161 -1187 -1213 -1239 -1216 -1242 -1219 -1245
## [782] -1271 -1297 -1274 -1300 -1277 -1254 -1280 -1306 -1332 -1358 -1335
## [793] -1312 -1289 -1315 -1292 -1318 -1295 -1321 -1298 -1275 -1301 -1327
## [804] -1353 -1330 -1356 -1382 -1359 -1385 -1362 -1339 -1316 -1342 -1319
## [815] -1296 -1322 -1348 -1325 -1302 -1279 -1256 -1282 -1308 -1285 -1262
## [826] -1288 -1265 -1242 -1268 -1245 -1222 -1199 -1176 -1153 -1130 -1156
## [837] -1133 -1110 -1136 -1113 -1139 -1165 -1142 -1168 -1145 -1122 -1099
## [848] -1076 -1053 -1030 -1007 -1033 -1059 -1085 -1111 -1137 -1114 -1091
## [859] -1068 -1094 -1071 -1048 -1074 -1051 -1028 -1054 -1031 -1008 -985
## [870] -962 -988 -1014 -1040 -1066 -1092 -1118 -1144 -1121 -1098 -1075
## [881] -1101 -1078 -1055 -1081 -1107 -1084 -1110 -1136 -1113 -1090 -1116
## [892] -1142 -1119 -1096 -1073 -1050 -1076 -1102 -1128 -1105 -1131 -1157
## [903] -1183 -1160 -1186 -1212 -1189 -1166 -1143 -1169 -1146 -1123 -1100
## [914] -1126 -1103 -1129 -1106 -1083 -1060 -1037 -1014 -991 -968 -945
## [925] -922 -948 -974 -1000 -1026 -1003 -980 -957 -983 -1009 -986
## [936] -1012 -989 -966 -992 -969 -995 -972 -998 -1024 -1050 -1027
## [947] -1053 -1030 -1007 -984 -961 -938 -915 -941 -967 -944 -970
## [958] -996 -973 -999 -976 -953 -979 -956 -933 -910 -936 -962
## [969] -939 -916 -942 -919 -896 -922 -948 -974 -1000 -977 -954
## [980] -931 -908 -934 -911 -937 -963 -940 -917 -894 -920 -946
## [991] -923 -949 -926 -903Detecting Liar: \(P(DL) = 0.59\)
Detecting Truth: \(P(DT) = 0.9\)
Probability of Someone Lying: \(P(L) = 0.2\)
Prob of being a liar after being detected: \(P(DL) \times P(L) = 0.59*0.2 = .118\)
So, \(\frac{.118}{.198} = 0.6\)
Prob of being a truth-teller after being detected: \(P(DT)\times P(1-L) = .9*.8 = .72\)
So, \(\frac{.72}{.802} = 0.9\)
Prob of being identified as a liar:
Prob of NOT detecting the truth in a random person: \(P(1-L)\times P(1-DT) = .1 * .8 = 0.08\)
Total probability of detecting a liar in a radom person: \(P(DL)\times P(L) + P(1-L)\times P(1-DT) = .59*.2+.1*.8 = 0.198\)
So, \(\frac{.08}{.198} = .4\)
Probability of being a liar or being identified as a liar: \(.4 + .198 = 0.6\)