Given the numerous discussions around vaccines and clinical trials that we have had this year, we would like you to get a deeper understanding of what some of these terms mean. We will look at a simplified version of vaccine trial information and try to understand what vaccine efficacy really means. There is much more that can be extrapolated from vaccine trial data, but we will be limiting ourselves to a small subset. Table 1 is a simplified version of Table 3 from the paper that discusses the efficacy of BNT162b2, the Pfizer-made vaccine for COVID-19: Efficacy Endpoint Number of COVID Cases Number of COVID Cases Vaccine Efficacy, % Subgroup in Vaccine Group (N=17,397) in Placebo Group (N=17,498) Overall 8 162 95.0 Age Group 16-55 years 5 (N=9897) 114 (N=9955) 95.6 >55 years 3 (N=7500) 48 (N=7543) 93.7 >65 years 1 (N=3848) 19 (N=3880) 94.7 >75 years 0 (N=774) 5 (N=785) 100.0(a) lfyou don't read the table carefully, you might come to a strange (and incorrect) conclusion: you are more likely to get sick if you are younger. From the table, the probability of getling sick if you were between 16 and 5 5 is w (9, 897 + 9,955) (3+1+0+48+19+ 15) (7, 500 + 3,848 + 774 + 7, 543 + 3, 880 + 785) m 0.006, while the probability of getting sick if you in the other age groups is 2: 0.003. What is the issue with this calculation? (b) lfyou pick three distinct participants, what is the probability that they all fall in the same age range? Here we define distinct age ranges to be 16-55, 55-65, 65-75, and >75. (c) What is the probability that at least two of three randomly chosen participants got the vaccine? (d) What is the probability that at least two of three randomly chosen participants got the vaccine AND that they all fell into the same age group