In an experiment to test the effectiveness of the latest roach trap, the "Roach Resort,"100roaches were placed in the vicinity of the trap and left there for an hour. At the end of the hour, it was observed that50of them had "checked in," while the rest were still scurrying around. (Remember that "once a roach checks in, it never checks out." Let 1 = not checked in, and 2 = checked in.)

(a) Set up the transition matrixPfor the system with decimal entries.

CalculateP².

CalculateP³.

(b) If a roach begins outside the "Resort," what is the probability of it "checking in" by the end of 1 hour? 2 hours? 3 hours?1 hour2 hours3 hours

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

Eventually, all the roaches will have checked in.

Eventually, all the roaches will not have checked in.

There is not enough information.