A new restaurant in my hometown gets most of its income from a few important clients. I want to model the probability that a client comes to my restaurant.
Let P(i) be the probability that a client shows up on day i as a function of the total number of clients who visited the restaurant the previous day, n. Assume each client’s decision to go to the restaurant is independent.
Simple case:
- There are 3 clients in the hometown.
- The probability function is defined as f(n) = 1 – n/3 when n = 1, 2, or 3.
- f(0) = 0.
Question 1: What is the probability of exactly one client in the restaurant if there were 2 customers at the restaurant the previous day?
Question 2: Calculate the expected number of days until no people show up.
Question 3: Now instead of f(0) = 0, suppose f(0) is split evenly between 0 people showing up and 1 person showing up. What is the expected number of clients dining in my restaurant far in the future?
This problem is a small-state probability and Markov chain expectation question. With only 3 clients, each client acts independently, and the next-day arrival probability depends only on how many visited the previous day, so the process can be modeled with states 0, 1, 2, and 3. The first question uses f(2) = 1/3 directly to find the probability that exactly one client appears after a day with 2 customers. The second question asks for the expected time to reach 0 visitors, which can be solved by treating state 0 as absorbing and setting up recurrence equations for the expected hitting time. The third question changes the 0-state behavior so that the chain no longer necessarily gets absorbed, and the long-run expected number of dining clients is found from the stationary distribution or steady-state equations.
