The number of people arriving for treatment at an emergency room can be modeled by a Poisson process with a rate parameter of four per hour. (a) What is the probability that exactly three arrivals occur during a particular hour? (Round your answer to three decimal places.) (b) What is the probability that at least three people arrive during a particular hour? (Round your answer to three decimal places.) (c) How many people do you expect to arrive during a 30-min period? people

Answer: a) 0.1954

b) 0.3908

c) 8

Step-by-step explanation:

For Poisson distribution,

P (x=r) = (e^-u x u^r) / r!

Where u = mean

x = r = number of arrivals

From the information given,

mean, u = 4 (a rate parameter of four per hour)

a) The probability that exactly three arrivals occur during a particular hour is

P (x=3) = (e^-4 x 4^3) / 3!

P (x=3) = 1.1722/6 = 0.1954

(b) The probability that at least three people arrive during a particular hour

= P (x greater than or equal to 3)

= P (x=3) + P (x=4)

P (x=3) = (e^-4 x 4^3) / 3!

P (x=3) = 1.17220088888/6 = 0.19536681481 = 0.1954

P (x = 4) = (e^-4 x 4^4) / 4!

P (x=4) = 4.68880355552/24 = 0.19536681481 = 0.1954

P (x greater than or equal to 3) =

0.1954 + 0.1954 = 0.3908

c) number of people expected to arrive during a 30-min period would be 4*2 = 8

This is because 4 arrive in an hour, so eight will arrive in 0.5 hours