With n = 8 and p = 0.9, find P (at least 6).

Do the following: (a) Find the indicated binomial probability by using Table A-1 in Appendix A. (b) If np ≥ 5 and nq ≥ 5, also estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution; if np < 5 or nq < 5, then state that the normal approximation is not suitable.

Table A-1 Binomial Probabilities

.01 05 10 20 30 .40 50 .60 .70 80 .90 .95 .99 2 980 .902 .810 .640 .490 360 250 160 .090 040 .010 .002 .020 .095 180 320 .420 .480 500 .480 .420 320 180 .095 020 .002 .010 .040 .090 160 250 360 .490 640 .810 .902 980 .970 .857 729 512 .343 216 .125 .064 .027 .008 L01 O+ O+ .029 135 243 384 ,441 432 375 .288 189 .096 .027 .007 O+ 1 .007 .027 .096 189 288 .375 .432 .441 384 243 135 .029 2 O+ .01 008 .027 064 .125 .216 .343 .512 729 .857 .970 4 .961 .815 .656 410 .240 130 .062 .026 .008 .002 1 .039 171 292 410 412 346 250 154 .076 .026 .004 .01 .014 .049 154 265 346 375 346 265 154 .049 .014 .01 .004 .026 .076 154 250 .346 .412 410 .292 171 .039 O+ O+ .002 .008 .026 062 130 240 410 .656 815 961 4 .951 774 .590 328 .168 .078 .031 .010 002 O+ O+ .048 204 328 410 360 259 156 .077 .028 .006 O+ O+ .001 .021 .073 .205 .309 .346 .312 .230 132 .051 .008 .001 O+ 2 3 O+ .01 .008 .051 132 230 312 .346 .309 .205 .073 .021 01 3 0+ O+ .006 .028 .077 156 259 360 410 328 .204 .048 4 O+ O+ 002 .010 .031 .078 168 .328 .590 .774 .951 5. 6. .941 .735 531 .262 118 047 016 .004 .001 1 057 232 354 .393 .303 .187 .094 .037 .010 .002 2 .01 .031 .098 246 324 311 234 138 060 .015 01 O+ .002 .015 .082 185 276 312 276 185 .082 .015 .002 O+ 0+ 0+ .01 .015 .060 138 234 311 324 246 .098 .031 .001 4 O+ .002 010 .037 .094 187 303 393 354 232 .057 01 .004 .016 .047 118 .262 531 735 941 6. 7 932 .698 .478 210 082 .028 .009 002 O+ 0+ O+ O+ 0+ 066 .257 372 367 247 .131 .055 .017 .004 0+ 1 2 002 .041 124 275 318 261 164 .077 .025 .004 O+ 2 3. O+ .004 .023 115 227 290 273 194 097 .029 003 0+ O+ .003 .029 097 194 .273 290 227 .115 023 .004 4 O+ 0+ .004 .025 077 164 261 318 275 124 .041 .002 5 6 0+ 0+ .004 .017 .055 131 .247 367 372 257 .066 6 7 0+ .002 .008 028 .082 210 .478 .698 .932 7 8 .923 .663 .430 168 058 .017 .004 .01 0+ O+ .075 .279 .383 336 198 .090 .031 .008 .01 O+ O+ 003 .051 149 294 .296 .209 .109 041 010 .01 O+ .005 .033 147 .254 279 219 124 047 009 3. .005 046 136 232 273 232 136 046 .005 5 O+ .009 047 124 219 279 .254 147 .033 .005 0+ 6 O+ .001 .010 .041 109 209 296 294 149 .051 .003 6 O+ .001 .008 .031 .090 198 336 383 279 .075 0+ .01 .004 .017 .058 168 430 .663 923 8 00