Question

5. The CDC receives reports of 7.7 cases of typhoid fever per week, on average, from all over the United States. Typhoid fever is

a life-threatening bacterial illness caused by Salmonella typhi and is transmitted by ingestion of contaminated food or drinks.

Use the Poisson distribution to find the probability of…

a) Getting a weekly report of no more than three cases of typhoid fever

b) Getting a report of more than 7 cases of typhoid fever in one week

Answer 1

a)
Here, λ = 7.7 and x = 3
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X <= 3).
P(X <= 3) = (7.7^0 * e^-7.7/0!) + (7.7^1 * e^-7.7/1!) + (7.7^2 * e^-7.7/2!) + (7.7^3 * e^-7.7/3!)
P(X <= 3) = 0.0005 + 0.0035 + 0.0134 + 0.0345
P(X <= 3) = 0.0519

b)
We need to calculate P(X > 7) = 1 - P(X <= 7).
P(X > 7) = 1 - (7.7^0 * e^-7.7/0!) + (7.7^1 * e^-7.7/1!) + (7.7^2 * e^-7.7/2!) + (7.7^3 * e^-7.7/3!) + (7.7^4 * e^-7.7/4!) + (7.7^5 * e^-7.7/5!) + (7.7^6 * e^-7.7/6!) + (7.7^7 * e^-7.7/7!)

P(X > 7) = 1 - (0.0005 + 0.0035 + 0.0134 + 0.0345 + 0.0663 + 0.1021 + 0.1311 + 0.1442)

P(X > 7) = 1 - 0.4956

= 0.5044