Suppose we are interested in analyzing the market share and customer loyalty for Murphy's Foodliner and Catalina's Supermarket, the only two grocery stores in a small town. We focus on the sequence of shopping trips of one customer and assume that the customer makes one shopping trip each week to either Murphy's Foodliner or Catalina's Supermarket, but not both. The data shows that of all customers who shopped at Murphy's in a given week, 90% shopped at Murphy's the following week while 10% switched to Catalina's. The data shows that of all customers who shopped at Catalina's in a given week, 80% shopped at Catalina's the following week while 20% switched to Murphy's. Suppose that we are considering the Markov process associated with the shopping trips of one customer, but we do not know where the customer shopped during the last week. Thus, we might assume a 0.5 probability that the customer shopped at Murphy's (State 1) and a 0.5 probability that the customer shopped at Catalina's (State 2) at period 0; that is, ₁ (0) = 0.5 and ₂(0) = 0.5. Given these initial state probabilities, develop a table showing the probability of each state in future periods. (Round your answers to three decimal places.) 0 1 2 3 4 5 6 7 8 9 10 π₁ (n) 0.5 State Probability #₂(n) 0.5 What are the long-run probabilities of each state? (Enter your probabilities as fractions.) Murphy's π₁ = Catalina's #₂ =
Suppose we are interested in analyzing the market share and customer loyalty for Murphy's Foodliner and Catalina's Supermarket, the only two grocery stores in a small town. We focus on the sequence of shopping trips of one customer and assume that the customer makes one shopping trip each week to either Murphy's Foodliner or Catalina's Supermarket, but not both. The data shows that of all customers who shopped at Murphy's in a given week, 90% shopped at Murphy's the following week while 10% switched to Catalina's. The data shows that of all customers who shopped at Catalina's in a given week, 80% shopped at Catalina's the following week while 20% switched to Murphy's. Suppose that we are considering the Markov process associated with the shopping trips of one customer, but we do not know where the customer shopped during the last week. Thus, we might assume a 0.5 probability that the customer shopped at Murphy's (State 1) and a 0.5 probability that the customer shopped at Catalina's (State 2) at period 0; that is, ₁ (0) = 0.5 and ₂(0) = 0.5. Given these initial state probabilities, develop a table showing the probability of each state in future periods. (Round your answers to three decimal places.) 0 1 2 3 4 5 6 7 8 9 10 π₁ (n) 0.5 State Probability #₂(n) 0.5 What are the long-run probabilities of each state? (Enter your probabilities as fractions.) Murphy's π₁ = Catalina's #₂ =
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:Suppose we are interested in analyzing the market share and customer loyalty for Murphy's Foodliner and Catalina's Supermarket, the only two grocery stores in a small town. We focus on the sequence of shopping trips of one customer and assume that the
customer makes one shopping trip each week to either Murphy's Foodliner or Catalina's Supermarket, but not both. The data shows that of all customers who shopped at Murphy's in a given week, 90% shopped at Murphy's the following week while 10%
switched to Catalina's. The data shows that of all customers who shopped at Catalina's in a given week, 80% shopped at Catalina's the following week while 20% switched to Murphy's. Suppose that we are considering the Markov process associated with the
shopping trips of one customer, but we do not know where the customer shopped during the last week. Thus, we might assume a 0.5 probability that the customer shopped at Murphy's (State 1) and a 0.5 probability that the customer shopped at Catalina's
(State 2) at period 0; that is, ₁ (0) = 0.5 and ₂(0) = 0.5.
Given these initial state probabilities, develop a table showing the probability of each state in future periods. (Round your answers to three decimal places.)
0
1
2
3
4
5
6
7
8
9
10
π₁(n)
π1
0.5
π2
What are the long-run probabilities of each state? (Enter your probabilities as fractions.)
Murphy's
Catalina's
=
State
Probability
=
π₂ (n)
0.5
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