Final answer:
The probability that at least one child does not develop an immune response to measles after receiving the aerosolized vaccine is 0.9999, while the probability with subcutaneous injection is 0.989.
Step-by-step explanation:
The probability that at least one child does not develop an immune response to measles after receiving the aerosolized vaccine can be calculated using complementary probabilities. If we assume that the probability of a child developing an immune response to the aerosolized vaccine is 0.84, then the probability that a child does not develop an immune response is 1 - 0.84 = 0.16. Since the researchers drew samples independently, we can calculate the probability that at least one child does not develop an immune response by using the complement of the probability that all children do develop an immune response. In this case, the probability that at least one child does not develop an immune response to the aerosolized vaccine is 1 - (0.16)^{50} = 0.9999.
Similarly, if all children were vaccinated using the subcutaneous injection of the vaccine, we can use the same approach to calculate the probability that at least one child does not develop an immune response. If the probability of a child developing an immune response to the subcutaneous injection is 0.86, then the probability that a child does not develop an immune response is 1 - 0.86 = 0.14. Therefore, the probability that at least one child does not develop an immune response to the subcutaneous injection is 1 - (0.14)^{50} = 0.989.