Chapter 8.7, Problem 39E

Pest Control In an experiment to test the effectiveness of the latest roach trap, the “Roach Resort,” 50 roaches were placed in the vicinity of the trap and left there for an hour. At the end of the hour, it was observed that 30 of the roaches had “checked in,” while the rest were still scurrying around. (Remember that “once a roach checks in, it never checks out.”)

a. Set up the transition matrix P for the system with decimal entries, and calculate $P^2\text{ and }{P}^3$.

b. If a roach begins outside the “Resort,” what is the probability of its “checking in” by the end of 1 hour? 2 hours? 3 hours?

c. What do you expect to be the long-term impact on the number of roaches? [HinT: See Example 5.]