Question

At the beginning of the term, students sign up for advising appointments at a Poisson rate of one student every 30 minutes. If 70% of the appointments are in-person, and the rest are virtual,

(a) What is the probability that at least one in-person appointment is requested during a 3-hour period?

Answer 1

The probability of at least one in-person appointment in a 3-hour period, when appointments are on a Poisson rate, is calculated by finding the complement of the probability of no in-person appointments during the given time frame.

Step-by-step explanation:

To find the probability of at least one in-person appointment being requested during a 3-hour period, we start by figuring out the rate of in-person appointments. Given that appointments are made at a Poisson rate of one every 30 minutes, there are 6 appointments in a 3-hour period. Since 70% of these are in-person, we have an average rate (λ) of in-person appointments of 0.7 * 6 = 4.2 in 3 hours.

The probability of getting at least one in-person appointment in 3 hours is the complement of having no in-person appointments. Thus, we calculate P(at least one in-person) = 1 - P(no in-person appointments). The probability of no in-person appointments in a time interval in a Poisson distribution is given by
P(X=0) = e^{-λ} where λ is the average rate.

Therefore, P(no in-person appointments) = e^{-4.2}. Finally, to find the probability of at least one in-person appointment, we subtract this probability from 1: P(at least one in-person) = 1 - e^{-4.2}.