Question

# 5. The CDC receives reports of 7.7 cases of typhoid fever per week, on average, from...

 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

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

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