Answer:
Probability that in a randomly selected office hour in the 10:30 am time slot exactly two students will arrive is 0.224.
Step-by-step explanation:
We are given that a statistics professor finds that when he schedules an office hour at the 10:30 am, time slot, an average of three students arrive.
Let X = number of students arriving at the 10:30 AM time slot.
<em>The above situation can be represented through Poisson distribution because a random variable representing the occurrence or arrival of events in a fixed period of time is known as Poisson random variables. </em>
The probability distribution function of Poisson distribution is given by;

where,
= arrival rate of students
As we know that the mean of Poisson distribution = 
So, X ~ Poisson(
)
Thus, the probability that in a randomly selected office hour in the 10:30 am time slot exactly two students will arrive is given by = P(X = 2)
P(X = 2) =
=
= 0.224
<em>Therefore, probability that in a randomly selected office hour in the 10:30 am time slot exactly two students will arrive is 0.224.</em>