Answer:
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.