Answer:
(a) the mean number for new infections is equal to 1
(b) the probability that no more than 2 of the 20 children develop infections is 0.9245
Step-by-step explanation:
If all 20 have been vaccinated, the number of new infection follows a binomial distribution, so, the probability that x children develop infections is calculated as:


where n is the number of children and p is the probability to develop the infection given that all 20 have been vaccinated. Additionally, the mean number E(x) of new infections is equal to:

Now, taking into account that all children are vaccinated, we can replace n by 20 and p by 0.05, so the mean number of new infections is:

Then, the probability that no more than 2 of the 20 children develop infections is calculated as:

where p(x) is:

So, p(0), p(1) and p(2) are equal to:

Finally,
is equal to:
