SECTION 4.2 Binomial Distributions 207 Finding binomial probabilities with the binomial probability formula can be a tedious process. To make this process easier, you can use a binomial probability table. Table 2 in Appendix B lists the binomial probabilities for selected values of n and p. Finding a Binomial Probability Using a Table About 5% of employees (ages 16 years and older) in the United States commute to their jobs by using public transportation (excluding taxicabs). You randomly select eight workers. What is the probability that exactly three of them use public transportation to get to work? Use a table to find the probability. (Source: American Community Survey) SOLUTION A portion of Table 2 in Appendix B is shown here. Using the distribution for n = 8 and p = 0.05, you can find the probability that x = 3, as shown by the highlighted areas in the table. 8 0 .923 .663 .430 .272 .168 .100 .058 .032 .017 .008 .004 .002 .001 1 .075 .279 .383 .385 .336 .267 .198 .137 .090 .055 .031 .016 .008 2 .003 .051 .149 .238 .294 .311 .296 .259 .209 .157 .109 .070 .041 3 .000 .005 .033 .084 .147 .208 .254 .279 .279 .257 .219 .172 .124 4 .000 .000 .005 .018 .046 .087 .136 .188 .232 .263 .273 .263 .232 5 .000 .000 .000 .003 .009 .023 .047 .081 .124 .172 .219 .257 .279 6 .000 .000 .000 .000 .001 .004 .010 .022 .041 .070 .109 .157 .209 7 .000 .000 .000 .000 .000 .000 .001 .003 .008 .016 .031 .055 .090 8 .000 .000 .000 .000 .000 .000 .000 .000 .001 .002 .004 .008 .017 p n x .01 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 2 0 .980 .902 .810 .723 .640 .563 .490 .423 .360 .303 .250 .203 .160 1 .020 .095 .180 .255 .320 .375 .420 .455 .480 .495 .500 .495 .480 2 .000 .002 .010 .023 .040 .063 .090 .123 .160 .203 .250 .303 .360 3 0 .970 .857 .729 .614 .512 .422 .343 .275 .216 .166 .125 .091 .064 1 .029 .135 .243 .325 .384 .422 .441 .444 .432 .408 .375 .334 .288 2 .000 .007 .027 .057 .096 .141 .189 .239 .288 .334 .375 .408 .432 3 .000 .000 .001 .003 .008 .016 .027 .043 .064 .091 .125 .166 .216 According to the table, the probability is 0.005. You can check this result using technology. As shown at the right using Minitab, the probability is 0.0054165. After rounding to three decimal places, the probability is 0.005, which is the same value found using the table. Interpretation So, the probability that exactly three of the eight employees use public transportation to get to work is 0.005. Because 0.005 is less than 0.05, this can be considered an unusual event. TRY IT YOURSELF 6 About 85% of employees (ages 16 years and older) in the United States commute to their jobs by driving a car, truck, or van. You randomly select six workers. What is the probability that exactly four of them drive a car, truck, or van to work? Use a table to find the probability. (Source: American Community Survey) Answer: Page A38 MINITAB Probability Density Function Binomial with n = 8 and p = 0.05 x P(X = x) 3 0.0054165 To explore this topic further, see Activity 4.2 on page 214. 4.2 EXAMPLE 6
RkJQdWJsaXNoZXIy NjM5ODQ=