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

a) 0.195 = 19.5% probability that exactly three arrivals occur during a particular hour.

b) 0.762 = 76.2% probability that at least three people arrive during a particular hour

c) 2 people.

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

In which

x is the number of sucesses

e = 2.71828 is the Euler number

is the mean in the given interval.

Poisson process with a rate parameter of four per hour.

This means that .

(a) What is the probability that exactly three arrivals occur during a particular hour?

This is P(X = 3). So

0.195 = 19.5% probability that exactly three arrivals occur during a particular hour.

(b) What is the probability that at least three people arrive during a particular hour?

Either less than three people arrive during a particular hour, or at least three does. The sum of the probabilities of these outcomes is 1. So

We want , in which

. So

So

0.762 = 76.2% probability that at least three people arrive during a particular hour.

c) How many people do you expect to arrive during a 30-min period?

One hour, 4 people

Half an hour = 30 min = 4/2 = 2 people.