Question

Whooping cough (pertussis) is a highly contagious bacterial infection that was a major cuase of childhood deaths before the development of vaccines. About 80%80% of unvaccinated children who are exposed to whooping cough will develop the infection, as opposed to only about 5%5% of vaccinated children.

A group of 18 children at a nursery school are exposed to whooping cough by playing with an infected child. If all 18 children have been vaccinated, then what is the probability that no more than 4 of the 18 children develop infections?

Answer 1

It is given that 5% of vaccinated children will develop an infection i.e p = 0.05

This is a binomial distribution as it has just two outcomes - infection or no infection.

n = 18 students
We need to find: P(X <= 4)

From the binomial formula,

P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

X	p(x)
0	0.397
1	0.376
2	0.168
3	0.047
4	0.009
Total	0.998

Hence, the probability that no more than 4 of the 18 children will develop infections is 0.998