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Question 2: A café in a bustling city center experiences customer arrival according to a Poisson process with a rate of 2 customers per hour. Let $X(t)$ represent the number of customers that have arrived by time t. The café manager is interested in analyzing customer flow to optimize staffing and seating arrangements. a. What is the probability that exactly two customers have arrived by the end of the first hour? b. What is the probability that exactly two customers have arrived by the end of the first hour, given that six customers have arrived by the end of the third hour? c. What is the probability of having six customers in total by the end of the third hour, given that two customers arrived in the first hour? d. What is the probability that the first customer arrives within the first 45 minutes of opening? e. If no customer has arrived in the first 45 minutes, what is the probability that the first customer arrives between 45 and 75 minutes after opening? f. What is the probability that the time interval between the arrival of the third and fourth customers exceeds 30 minutes? g. Given that the second customer arrives exactly 30 minutes after opening, what is the probability that a total of four customers have arrived by the end of the first hour?

          Question 2: A café in a bustling city center experiences customer arrival according to a Poisson process with a rate of 2 customers per hour. Let $X(t)$ represent the number of customers that have arrived by time t. The café manager is interested in analyzing customer flow to optimize staffing and seating arrangements.
a. What is the probability that exactly two customers have arrived by the end of the first hour?
b. What is the probability that exactly two customers have arrived by the end of the first hour, given that six customers have arrived by the end of the third hour?
c. What is the probability of having six customers in total by the end of the third hour, given that two customers arrived in the first hour?
d. What is the probability that the first customer arrives within the first 45 minutes of opening?
e. If no customer has arrived in the first 45 minutes, what is the probability that the first customer arrives between 45 and 75 minutes after opening?
f. What is the probability that the time interval between the arrival of the third and fourth customers exceeds 30 minutes?
g. Given that the second customer arrives exactly 30 minutes after opening, what is the probability that a total of four customers have arrived by the end of the first hour?

Question 2: A café in a bustling city center experiences customer arrival according to a Poisson process with a rate of 2 customers per hour. Let X(t) represent the number of customers that have arrived by time t. The café manager is interested in analyzing customer flow to optimize staffing and seating arrangements.
a. What is the probability that exactly two customers have arrived by the end of the first hour?
b. What is the probability that exactly two customers have arrived by the end of the first hour, given that six customers have arrived by the end of the third hour?
c. What is the probability of having six customers in total by the end of the third hour, given that two customers arrived in the first hour?
d. What is the probability that the first customer arrives within the first 45 minutes of opening?
e. If no customer has arrived in the first 45 minutes, what is the probability that the first customer arrives between 45 and 75 minutes after opening?
f. What is the probability that the time interval between the arrival of the third and fourth customers exceeds 30 minutes?
g. Given that the second customer arrives exactly 30 minutes after opening, what is the probability that a total of four customers have arrived by the end of the first hour?

Added by Daniel G.

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