Whooping cough (pertussis) is a highly contagious bacterial infection that was a

Question

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

A group of 30 children at a nursery school are exposed to whooping cough by playing with an infected child.

(a) If all 30 have been vaccinated, what is the mean number of new infections? (Enter your answer to two decimal places.)
_______new infections

What is the probability that no more than 3 of the 30 children develop infections, if all 30 have been vaccinated? (Round your answer to four decimal places.)


(b) If none of the 30 have been vaccinated, what is the mean number of new infections? (Round your answer to one decimal place.)
______new infections

What is the probability that 28 or more of the 30 children develop infections, if none of the 30 have been vaccinated? (Round your answer to four decimal places.)

Explanation / Answer

A.1. 30 x 0.05 = 1.5

    2. This is a case of binomial probability distribution.

        P (X = r) = nCr pr (1-p)n-r
Where, Combination nCr = ( n! / (n- r)! ) / r!; n = Number of Events; r = Number of Success; p = Probability of Success

       P ( X < 3) = P (X =0) +P (X =1) +P (X =2)+P (X =3) = 0.9392.       

B.1. 30 x 0.80 = 24.

    2. P ( X < 28) = 1 - P ( X < 2) = 1 - [ P (X =0) +P (X =1) +P (X =2)] = 0.9895

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