An article in the daily newspaper Dagens Nyheter on 12 August 2020 explained why it is important to test a new vaccine (or drug) in order to avoid side effects as much as possible. "What are the risks...

Extracted text: An article in the daily newspaper Dagens Nyheter on 12 August 2020 explained why it is important to test a new vaccine (or drug) in order to avoid side effects as much as possible. "What are the risks of skipping important research steps? The whole world is hoping for a vaccine against covid-19, and more than 150 possible vaccines are under development. It has gone faster than ever before, but a vaccine still needs to be tested on humans in several different stages before it can be approved for use. The different research steps are called phase 1, phase 2 and phase 3. Phase 3 is the last step before an approval and then tests how safe and effective the vaccine is on a large number of people. The Modern vaccine candidate is tested, for example, on 30,000 people. Although the previous research phases should generally detect side effects, there may be more unusual, but also serious, side effects that are not detected until phase 3 studies. It is a step that cannot be skipped. By approving and using an inadequately tested vaccine, Russia is taking a very big risk, not just for its own population. Should the vaccine prove to have serious side effects, it could also discourage people around the world from being vaccinated with another vaccine, which has been shown to be effective and safe in studies. There is also a possibility that the vaccine does not provide sufficient protection and then the spread of infection risks is increasing if people behave as if they are protected when in fact they are not." A side effect that affects an average of 1 person in 10,000 people is "very rare". a) Assume that a vaccine candidate has a side effect that is very rare and that a Phase 3 study is being conducted with 30,000 subjects. Calculate the probability that the study produces at least one person with a very rare side effect. A side effect that affects an average of 1 person in 100,000 people is "extremely rare". b) Assume that a vaccine candidate has a side effect that is extremely rare and that a Phase 3 study is being conducted with 30,000 subjects. Calculate the probability that the study produces at least one person with an extremely rare side effect. c) Assume that 2 million people are vaccinated and that the vaccine has an extremely rare side effect. Find two constants – a and b – such that the probability is 99 % that the number of people affected by the extremely rare side effect is in the interval (a, b).