Solutions:

1. If a fraction f take the flu shot, then the death toll is 5f + 10(1-f)(1-f) = 5f + 10f2 - 20f + 10. That is, 10f2 - 15f + 10. This value is minimized when 20f = 15 or f = 0.75. At that point, we'd have a death toll of  only (5 * 0.75) + (2.5 * 0.25) = 4.375%. Notice however, that those who don't take the vaccine have a probability of only 2.5 percent of dying whereas the ones that do take the flu shot have a 5 percent chance of dying.

2. Exactly half (or maybe one less than half given the bias toward inaction). Here is why. Take the ith person. If half have already taken the flu shot, then there is no reason to take it. Person i will die with a 5 percent probability if he or she takes the shot and will die with  a 5 percent probability or less if he or she doesn't take the shot. If less than half have taken it and there are just enough people left for half to take it, then person i will take it for sure, because if he or she doesn't, the chances of living will be less than if he or she took the shot. We call this the "must vaccinate" point.

3. The government wants f = 0.75 from the answer to 1. So if the initial perceived risk is R, each person will take then R(1-f) = R 0.25 = 5%. So R = 20%. That is, if everyone believes the initial unvaccinated risk is 20 percent, then 75 percent of the people will take the vaccine.