(a) P = 0.2 0.8 0.8 0.2

,

v = [100   0]

Probability of going from state 2 to state 1 in two steps =      

Distribution vector after two steps is [   ]      

* Also assume that there is an inexhaustible supply of science fiction, Proust, and calculus books.

(a) At noon, the entire beach population of 1,200 was in the swimming area. How crowded was the jetty by 2 pm.?
people      

(b) What fraction of the people in the volleyball play area at noon will be at the jetty by 2 pm.?
at the jetty      

5. Malls Following is a plan of the "Petit Mall." As an potential store owner, you have been offered a lease at various store locations in the mall (the colored areas show the store locations), and are deciding which location to choose. Part of your decision will be based on a knowledge of shopper traffic in different parts of the mall (areas A through F).

During the first part of the morning hours, very few shoppers leave the mall through tghe exit door and the movie theater is closed. Further, shoppers tend to move randomly from one area to an adjacent area. For instance, a shopper in area B has a 1/3 chance of moving to A, C, or D, which a shopper in C is certain to move to B.

(a) If a shopper begins in location A when the mall opens, what is the probability of that shopper being found in location C after two changes of location (two time steps)?

(b) Calculate P², P³, P⁴, and P⁵ by hand. On average, what percentage of shoppers can be found where they started after 5 changes of location?

6. Sports Researchers have modeled the results of cricket games between England and Australia by the following Markov system, in which a time step corresponds to one game. (The states represent wins, losses and draws for England.) ^††

In the long term, England can expect to win of the time, to lose of the time, and to draw of the time.

†† Figures are rounded to one decimal place. Source: Derek Colwell, Brian Jones and Jack Gillet, "A Markov Chain in Cricket," the Mathematical Gazette, June, 1991.

7. Heating (Use of technology recommended. Try the the on-line matrix algebra tool for a direct computaiton, or the Pivot and Gauss-Jordan Tool to solve the associated system of equations.)
According to research by Gulf Oil Corporation, switching between oil, gas and electric heat in US homes followed the following pattern over a one-year period. ^Ý

What are the long term implications of these trends? (Round all percentages to two decimal places.)

Ý Source: Ali Ezzarti, "Forecasting Market Shares of Alternative Home Heating Units," Management Science, Vol. 21, no. 4, Dec. 1974. Copyright (c) 1974 by The Institute of Management Sciences.

(a) Write down the transition matrix P, and compute the steady state transition vector v.
(b) Is the given system regular?

10. In the Markov system represented by the following transition diagram, determine how long, on average, it will take to reach the absorbing state, assuming that the system starts in State 2.

10. Petit Mall This continues the above question on the "Petit Mall."

During the evening hours, a shopper in Area C has a 0.5 probability of leaving the mall through Area E, while a shopper in area D has a 0.5 probability of going to the movies. Since the mall will be closed by the time the movie ends, regard areas E and F as absorbing states.

(a) Calcuate the fundamental matrix Q .

(b) How many visits will a shopper entering the mall in Area A be expected to make to Area D?

(c) Which area or areas will be visited the most frequently by a shopper entering the mall at Area A?

(d) What fraction of shoppers in Area D will ultimately go to the movies?