5-3 Poisson Probability Distributions 237 c. Find the probability that in a single day, there are no births. Would 0 births in a single day be a significantly low number of births? 10. Murders In a recent year (365 days), there were 650 murders in Chicago. Find the mean number of murders per day, then use that result to find the probability that in a single day, there are no murders. Would 0 murders in a single day be a significantly low number of murders? 11. Radioactive Decay Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying cesium-137, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms; therefore 22,713 atoms decayed during 365 days. a. Find the mean number of radioactive atoms that decayed in a day. b. Find the probability that on a given day, exactly 50 radioactive atoms decayed. 12. Deaths from Horse Kicks A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-years included a total of 196 deaths. After finding the mean number of deaths per corps-year, find the probability that a randomly selected corps-year has the following numbers of deaths: (a) 0, (b) 1, (c) 2, (d) 3, (e) 4. The actual results consisted of these frequencies: 0 deaths (in 144 corps-years); 1 death (in 91 corps-years); 2 deaths (in 32 corps-years); 3 deaths (in 11 corps-years); 4 deaths (in 2 corps-years). Compare the actual results to those expected by using the Poisson probabilities. Does the Poisson distribution serve as a good tool for predicting the actual results? 13. World War II Bombs In analyzing hits by V-1 buzz bombs in World War II, South London was partitioned into 576 regions, each with an area of 0.25 km2. A total of 535 V-1 buzz bombs hit the combined area of 576 regions. a. Find the probability that a randomly selected region had exactly 2 hits. b. Among the 576 regions, find the expected number of regions with exactly 2 hits. c. How does the result from part (b) compare to this actual result: There were 93 regions that had exactly 2 hits? 14. Disease Cluster Neuroblastoma, a rare form of cancer, occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children. a. Assuming that neuroblastoma occurs as usual, find the mean number of cases in groups of 12,429 children. b. Using the unrounded mean from part (a), find the probability that the number of neuroblastoma cases in a group of 12,429 children is 0 or 1. c. What is the probability of more than one case of neuroblastoma? d. Does the cluster of four cases appear to be attributable to random chance? Why or why not? 15. Chocolate Chip Cookies In the production of chocolate chip cookies, we can consider each cookie to be the specified interval unit required for a Poisson distribution, and we can consider the variable x to be the number of chocolate chips in a cookie. a. Refer to Data Set 39 “Chocolate Chip Cookies,” and find the mean number of chocolate chips in the 34 Keebler cookies. b. Assume that the Poisson distribution applies. Find the probability that a Keebler cookie will have 26 chocolate chips. c. Find the expected number of Keebler cookies with 26 chocolate chips among 34 different Keebler cookies. Compare the result to the actual number of Keebler cookies with 26 chocolate chips. 16. Chocolate Chip Cookies Repeat Exercise 15 using 30 chocolate chips instead of 26 chocolate chips. In Data Set 39 “Chocolate Chip Cookies,” six of the Keebler cookies have 30 chocolate chips.
RkJQdWJsaXNoZXIy NjM5ODQ=