Consider a binomial distribution with 10 trials. Look at Table 2 in the Appendix showing binomial probabilities for various values of p, the probability of success on a single trial.
(a) For what value of p is the distribution symmetrical? What is the expected value of this distribution? Is the distribution centered over this value?
(b) For small values of p, is the distribution skewed right or left?
(c) For large values of p, is the distribution skewed right or left?

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This table shows the probability of r successes in n independent trials, each with probability of success p. n rO 01 05 .10 15 .20 25 30 35 40 45 .50 55 .60 65 70 75 8085 90 95 2 0 90 902 810 723 640 .563 490 423 360 303 250 203 160 123 090 063 040 023 0 .002 .000 002 010 023 040 063 090 123 .160 203 250 303 360 423 490 .563 640 723 810 .902 3 0 970 857 729 614 512 422 343 275 216 166 125 091 064 043 027 016 00B .003 001 000 1 .029 .135 .243 .422 .441 .444 .432 .408 .375 .334 .288 .239 .18g .141 .096 ·057 .027 .007 325 384 000 007 027 057 096 441 444 432408 375 334 239 189 444 441 422 384 325 41 189 239 .288 334 375 408 275 343 422 512 614 29 857 4 0 961 815 656 .522 410 316 240 179 130 .092 062 041 026 015 008 004 002 00 000 .000 1 .039 .171 .292 .368 .410 .422 .412 .384 .346-300 .250 .200 .154 .112 .076 .047 .026 .011 .004 .000 000 000 003 008 016 027 043 064 09 125 166 001 014 049 098 154 211 265 346 368 375 368 346 311 26 000 000 004 011 026 .047 076 11 076 047 026 01 004 000 11 .154 098 049 014 154 200 250 300 346 384 412 422 410 368 292 171 2 5 0 .951 .774 .590 .444 .328 .237 .168 .116 .078 .050 .031 .019 .010 .005-002-001 .000 .000 .000 .000 1-048 .204 .328 .392 .410 .396 360 .312 .259 .206 .156 .113 .077 .049 .028 .015 .006 .002 .000 .000 2 .001 .021 .073-138 .205 .264 309 .336 .346 .337 .312 .276 .230 .181 .132 .088 .051 .024 .008 .001 .000 .000 .000 .002 .006 .015 .028 .049 .077 .113 .156 .206-259-312-360 396 10 392 328 204 6 0 941 735 531 377 262 178 118 075 047 028 .016 008 .004 002 .001 000 000 000 .000 .000 1 .057 .232 .354 .399 .393 .356-303 .244 .187 .136 .0g4 .061-037 .020 .010 .004 .002 .000 .000 .000 .324 .328 .311 .278 .234 .186 .138 .095 .060 .033 .015 .006 .001 .000 3 000 002 015 042 082 132 185 236 276 303 312 303 276 .236 185 132 082 042 015 002 324 297 246 176 098 031 000 .000 000 000 .002 004 010 020 037 061 094 136 187 244 303 356 393 399 354 232 .000 .000 .000 .000 .000 .000 .001 .002 .004 .008 .016 028 .047 .075 118-178 262 .377 531 .735 .932 .698 .478 321 .210 .133 .082 .049 .028 .015 .008 .004 .002 .001 .000 .000 .000 .000 .000 .000 丨 .066 .257 .372 .396 .367 .311 .247 .185-131 .087 .055 .032 .017 .008 .004 .001 .000 .000 .000 .000 002 041 124 210 275 311 318 299 261 214 164 17 077 047 025 012 004 001 .000 000 .001 .031 .098 .176 .246 4 000 000 001 006 .033 060 095 38 186 234 .278 311 000 000 003 011 029 .058 097 144 .194 239 273 292 290 268 227 173 115 062 023 004 5 000 000 000 00 004 012 025 047 077 117 164 214 261 .299 318 3 25 210 124 041 6 .000 .000 .000 .000 .000 .00] .004 .008 .017 .032 .055 .087-131 .185 .247 311 .367 .396 .372 .257 7 000000 000 000 000 000 000 001 002 .004 008 015 028 049 082 133 210 3 478 698 n .01 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 8 0 923 663 430 272 168 100 058 032 017 .008 004 .002 001 000 000 000 .000 .000 000 000 1075 .279 383 385 336 267 198 137 090 055 03 016 008 00 001 000 .000 .000 .000 000 2 .003 05 149 .238 294 311 296 259 209 57109 070 041 022 010 004 .001 000 .000 000 3 .000 .005 .033 .084 .147.208 .254 .279 .279 .257 .219 .172 .124-081 .047 .023 .009 .003.000 .000 4 .000 000 005 018 046 087 136 188 232 263 273 263 .232 188 136 087 046 018 .005 000 .000 .000 .000 .003 .009.023 .047 .081 .124 ·172 219 .257 .279 .279 .254 .208 .147 .084.033 .005 .000 .000 .000 .000 .001 .004 .010 .022 .041 .070 .109 .157 .209 .259 .296 .311 .294 .238 .149 .051 7 .000 .000 .000 .000 .000.000 .001 .003 .008 .016 .031 .055 .090 .137 .198 .267 .336 .385 .383 .279 8 .000 .000 .000 .000 .000 .000 .000 .000 .001 .002 .004 .008 이7 .032 .058 .100 .168 .272 .430 .663 9 0 .914 .630 387 .232 .134 .075 .040 .021 .010 .005 .002 .001 .000 .000 .000 .000 .000 .000 .000 .000 1 .083 .299 387 .368 .302 .225 .156 .100 .060 .034-018 .008 .004 .001 .000 .000 .000 .000 .000 .000 2 003 063 172 260 302 300 267 216 161 070 041 021 010 004 .001 000 000 000 .000 3 000 008 045 107 176 234 267 272 25 212 164 116 074 042 021 009 003 001 000 .000 4 .000 .001 .007 .028-066 .117 .172 .219 .251 .260 .246 .213 .167 .118 .074 .039 .017 .005 .001 .000 5 .000 .000 .001 .005 .017 .039 .074 .118 .167 213 .246 .260 .251 .219-172 .117 .066 .028 .007 .001 6 .000 .000.000.001 .003 .009 .021 .042 .074 .116 .164 .212 .251 272 .267 .234 .176 .107 .045 .008 7 000 000 000 000 000 001 004 .010 021 04 070 111 161 216 267300 302 260 172 063 8 000 000 000 000 000 000 000 001 004 008 018 034 060 100 156 225 302 368 387 299 9 000 000 000 000 000 000 000 000 000 00 002 005 010 021 040075 134 232 387 630 10 0 904 .599 349 197 .107 056 028 014 006 003 0OT 000 000 000 000 000 000 000 000 000 1 .091 .315 387 .347 .268 .188 .121 .072 .040 .021 010 .004 .002 .000 .000 .000 .000 .000.000 .000 2 .004 .075 .194 .276 .302.282 .233 .176 .121 .076 .044 .023 .01 1 .004 .001 .000 .000 .000 .000 .000 3 000 010 057 130 .201 250 267 252 .215 166 117075 042 021 009 .003 001 .000 000 000 4 .000 .001 .01 1 .040 .088.146 .200 .238 .251 .238 .205 .160 .111 .069 .037 .016 .006 .001 .000 .000 5 .000 .000 .001 .008.026.058 .103 .154 .201-234 .246 .234 .201 .154 .103 .058 .026 .008 .001 .000 6 000 000 .000 .001 006 016 037 .069 111 60 .205 238 251 238 200 146 088 040 011 001 7 .000.000.000.000.001 .003 .009 .021 .042 .075-117 .166 .215 .252 .267 .250 .201 .130.057-010 8 000 000 .000 000 000 000 O0T 004 011 023 044 076 121 176 233 282 302 276 194 075 9 000 .000 000 000 000 000 000 000 00 004 010 02 040 02 12 .188 268 347 387 315 10 .000 .000 .000 .000 .000 .000 .000 .00, .000 .000 .001 .0 S -DOG DA -U28 ·056 137 34) 59g 11 0 .895 .569 314 .167 .086 .042 .020 .009 .004 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 1 099 329 384 325 236 155 093 052 027 013 005 002 001 000 000 .000 .000 000 000 000 2 005 087 213 287 295 258 200 140 089 05 027 013 005 002 001 000 .000 000 .000 .000 r O1 .10 15 20 25 30 35 40 45 50 55 60 65 7075 80 85 90 95 11 3 .000 .014 .071 .152.221 .258 .257.225 .177 .126 .081 .046 .023 010 .004 .001 .000 .000 .000 .000 4 000 00T 016 054 11 1220 243 236 206 161 113 070 038 017 006 002 .000 000 000 5 .000 .000 .002 013 .039 .080 .132.183 .221 .236 .226.193 .147 .09g D57.027 .010 .002 .000 .000 6. .000 000 .000 002 010 027 057 099 147 193226 236 221 183132 .080 039 013 002 .000 7 000 000 000 00D 002 006 017 038 070 113 161 206 236 243 220 172 111 054 016 001 8 .000 000 000 000 000 00 004 010 023 046 081 126 177 225 257 258 221 152 071 014 9 .000 .000 .000 .000 .000 .000 .001 .002 .005 013 .027.051 .089 .140 그00-258 .2g5 .287 .213 .087 10 .000 .000 .000 .000 .000 .000 .000 .000 .001 .002 .005.013 .027 .052 .093 .155 .236 .325 .384 .329 11 .000 000 000 000 .000 000 000 000 000 .000 000 .001 .004 .009 020 042 086 167 314 569 12 0 .886 .540 .282 .142 .069 .032 .014 .006 .002 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 107 .341-377 .301 .206 .127 .071.037 ,017 .0 3,003 00, O00000 ㎜000 ,000, mu 2 006 .099 230 292 283 232 168109 064 034 016 007 .002 001 .000 000 000 000 000 000 000 .01 7 .085 .172.236 .258 .240.195 .142 .092 D54.028 012 .005 00i .000 ,000 .00油 4 .000 .002 .021 .068.133 .194 .231.237 .213 ,170 121,076 .042 .038 002 00: .000 .00 5 .000 000 .004 019 053 103 158 204 .227 223 193 149 101 059 029 011 003 001 000 .000 6 000 000 000 004 016 040 .079 128 177 212 226 212 177 128 079 040 016 004 .000 .000 7 .000 000 000 001 003 011 029 059 101 149 193 223 227 204 158 .103 053 019 004 000 8 000 000 .000 000 001 002 008 020 042 076121 170 213 237 231 194 13 068 021 002 g .000 .000 .000 .000 .000 .000 .001 .005 이2 .028 .054.092 .142 .195 .240.258 .236 .172 .085 .017 10 .000 .000 .000 .000 .000 .000 .000 .001 .002 .007 .016.034 .064 、Og -G3 232 2 292 240 11 .000 000 000 000 000 000 000 .000 .000 .001 .003 008 O17 037 071 127 206 301 377 341 12 000 000 000 .000 000 000 000 000 000 000 .000 .001 002 006 014 032 069 142 .282 540 15 0 860 463 206 087 035 013 005 .002 000 .000 .000 000 000 .000 000 .000 .000 000 000 000 1 130 366 343 231 132 067 031 013 005 002 .000 000 .000 .000 000 000 .000 000 000 000 2 .009 .135 .267 .286 .231-156 .092.048 .022 .009 .003.001 .000 .000 .000 .000 .000 .000 .000 .000 3 000 03 129 218 250 225 170 111 063 .032 014 005 .002 000 .000 000 000 .000 000 000 4 000 005 043 116 188 225219 179 127 078 042 019 007 002 001 000 000 .000 000 000 5 000 00 010 045 103 165206 212 186 140 .092 051 .024 010 .003 001 000 .000 000 000 000 .000 .002 .013-043 .092 .147.191 .207 .191 .153.T05 .061 .00 .012 003 oon oo 7 000 000 000 003 014 039 081 132 177 201 196 165 118 07 035 013 003 .00 000 000 8 .000 000 .000 001 003 013 .035 07 118 165 196 201 177 132 081 039 014 003 000 000 9 .000 .000 .000.000.001 .003 .012.030 .061 .105 .153 .191 .207 .191 .147.092 .043 013-002 .000 10 000 000 .000 000 .000 001 003 010 024 051 092140 186 212 206 165 103 045 010 .001 nr .01 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1511 .000.000.000.000.000 .000 .001 .002 .007 .019 .042 .078 .127 .179 .219 .225 ·188-116 .043 .005 000 000 000 000 000 000 000 000 002 005 014 032 063 1T1 170 225 250 218 129 03 5 000 000 000 000 000 000 000 000 000001 003 009 022 048 092 156 231 286 .267135 .000 000 000 000 .000 000 .000 000 000 000 000 .002 005 013 031 .067 132 23T 343 366 15 000 000 000 000 000 000 000 000 000 000 000 000 000 .002 005 .013 035 087 .206 463 16 0 85 440 185 074 028 010 003 001 000 000 000 000 000 000 000 .000 000 000 .000 000 1 .138 .371-329 .210 .113 .053 .023 .009 .003 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 2 010 146 275 277 211 134 073 035 015 006 .002 001 000 000 .000 000 000 000 .000 000 3 000 036 142 229 246 .208 146 089 047 022 009 003 OOT 000 000 .000 000 000 000 000 000 006 051 131 200 .225 204 155 10 057 028 01 004 001 .000 .000 .000 .000 000 000 7 6 000 000 003 018 055 110 165 .198 198 168 122 075 039 017 006 001 000 .000 000 .000 7 .000 .000 .000 .005.020.052 .101-152 ·189 ·197 .175 .132 .084 .044 .0g .006 .001 .000 .000 .000 8 .000 000 000 .001 006 020 049 092 142 181 196 1 142 092 049 .020 006 .0DT .000 .00D 9 .000 .000 .000 .000 .001 .006 .019-044 .084-132 .175-197 .189-152 ,101 .052 .000.005 .000 .000 10 .000 .000 .000.000.000.001 .006 .017 .039 .075 ·122-168-198-198-165 .110 .055 .018 .003 .000 11 .000 .000 .000.000.000 .000 .001 .005 .014 .034 .067-112 .162 .201 .210 .180 .120 .056 .014 .001 12 000 .000 .000 000 000 000 000 .001 .004 028 .057 1OT 155 204 225 200 13 05 .006 13 000 000 .000 .000 000 000 000 000 .00 003 009 022 047 089 146 .208 246 229 142 036 14 .000 .000 .000.000.000 .000 .000 .000 .000 .001 002 .006 .015 .035 .073-134 .211 .277 .275 .146 15 .000 .000 .000 .000.000.000 .000 .000 .000 .000 .000 .001 .003 .009 .023 .053 .113 .210 .329 .371 16 000 000 000 000 000 000 000 000 000 000 000 000 000 .001 003 .010 028 074 185 440 2 20 0 .818 358 ·122 .039 .012 .003 .001 000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 1 .165 .377 ..270-137 .058 .021 .007 .002 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 2 .016-189 285 .229 .137 .067 .028 .010 .003 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 3 .001 .060 .190.243 .205 .134 .072 .032 012 .004 .001 .000 .000 .000 .000 .000 .000 .000.000 .000 000 013 090 182 218 190 130 074 035 014 005 001 000 000 000 000 .000 000 000 000 5 000 .002 032 103 175 202 179 127 075 036 015 005 001 000 000 000 .000 000 .000 000 6 000 000 009 045 109169 92 171 124 075 037 015 005 001 000 000 000 000 00 000 .000 .000 .002 .016 .055、1 12 .164 .184-166-122 .074 .037 .015 .005 .001 .000 . .000 .000, .000 8 000 000 000 005 022 061 114 6 180162 120 OT3 035014 004 .00 000 .000 000 000 .000 .000 .000 .001 .007 .027 .065 .116 .160 .177 .160 .Mg .07T ·034 .012 .003 .000 .000 .000 .000 n 『 .01 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80-85 .90 .95 2010 .000.000.000.000.002.010 .031 .069-117 .159-176 .159-117-069.031 .010 .002 .000 .000 .000 11 000 000 000 000 000 003 012 034 071 119 10 177 160 16 065 027 007 001 000 .000 12 000 000 000 000 .000 .001 004 014 035 073 120 162 10 1 114 061 022 005 000 000 13 .000.000.000.000.000.000 .001 .005 .015 .037 .074 .122 .166 .184 .164 .112 .055 .016 .002 .000 14 .000.000.000.000.000.000 .000 .001 .005 .015 .037 .075-124-171 .192 .169 .109 .045.009 .000 15 000 000 000 000 .000 000 000 000 00 005 015 036 075 127 179 202 175 103 032 002 16 .000.000.000.000.000 .000 .000 .000 .000 .001 .005 .014 .035 .074 ·130 .190 .218 .182.090 .013 17 .000 .000.000.000 .000.000 .000 .000 .000 .000 .001 .004 .012 .032 .072 ·134 ·205 .243 .190 .060 18 000 000 000 000000 .000 000 000 000 000 000 001 003 010 028 067 137229 285 189 19 000 000 000 .000 .000 000 000 000 000 000 000 000 000 .002007 021 058 137 270 377 20 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .001 .003 .012 .039 .122 .358