Question

A certain region of a country is, on average, hit by 8.5 hurricanes a year. (a) What is the probability that the region will be hit by fewer than 7 hurricanes in a given year? (b) What is the probability that the region will be hit by anywhere from 6 to 8 hurricanes in a given year? Click here to view the table of Poisson probability sums. (a) The probability that the region will be hit by fewer than 7 hurricanes in a given year is (Round to four decimal places as needed.) (b) The probability that the region will be hit by anywhere from 6 to 8 hurricanes in a given year is (Round to four decimal places as needed.)

Answer 1

Solution :

Given that ,

mean = $\mu$ = 8.5

Using poisson probability formula,

P(X = x) = (e^-$\mu$ * $\mu$ ^x ) / x!

a)

P(X < 7) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)

P(X < 7) = (e^-4.3 * 4.3⁰⁾ / 0! + (e^-4.3 * 4.3¹⁾ / 1! + (e^-4.3 * 4.3²⁾ / 2! + (e^-4.3 * 4.3³⁾ / 3! + (e^-4.3 * 4.3⁴⁾ / 4! + (e^-4.3 * 4.3⁵⁾ / 5! +(e^-4.3 * 4.3⁶⁾ / 6!

P(X < 7) = 0.0136+0.0583+0.1254+0.1798+0.1933+0.1662+0.1191

P(X < 7) = 0.8558

Probability = 0.8558

b)

P(6<=X <= 8) = P(X=6) + P(X=7) + P(X=8)

P(6<=X < =8) = (e^-4.3 * 4.3⁶⁾ / 6! + (e^-4.3 * 4.3⁷⁾ / 7! + (e^-4.3 * 4.3⁸⁾ / 8!

=0.1191+0.0732+0.0393= 0.2316

=0.2316

Probability = 0.2316