3.1 Review: basic concepts of probability modeling A certain town has exactly one policeman (Jones) and exactly one burglar (Elmer). The town is divided into two police beats, each of which may be considered a straight line of length l. Each night the policeman makes an equally likely choice between the two beats and then spends the whole night patrolling the selected beat. When Jones is on a beat, his position at any time is uniformly distributed over the length of that beat.
    Tonight elmer will start committing one burglary per night until he is apprehended. On any particular night, given that he has not already been caught, Elmer is twice as likely to burglarize the beat that Jones is not patrolling than the one that he is patrolling.
    Elmer's burglary position is uniformly distributed over the beat that he has selected and is independent of Jone's position, even if he and Jones happen to have selected the same beat. Assume that Jones' position remains constant throughout the duration of the burglary.
    Given that Elmer and Jones are exactly w units of length apart on the same beat at the time of the burglary, Jones will apprehend Elmer with probability P{A|w}, as shown in Figure P3. 1. Note that P{A|w} is a conditional probability, not a probability density function.

a. What is the probability that Elmer and Jones will both work on the same beat tonight?

b. Given that Elmer and Jones are on the same beat tonight, and also given that they are separated by a distance of more than l/4 units, what is the conditional probability that they are separated by a distance of more than l/2 units?

c. Given that Elmer and Jones are on the same beat tonight, determine the pdf f_w(w) for - < w <   , where W is the magnitude of the distance between them at the time of the burglary.

d. Given that Elmer has not as yet been caught and given that tonight he and Jones choose the same beat, show that P_A, the conditional probability that he will be apprehended tonight, is (d/l)[1 - (1/3)(d/l)]. Does this answer seem reasonable for d = 0 and d = l?

e. Determine the probability that Elmer is apprehended for the first time on the third night.

f. Given that Elmer has successfully completed exactly 10 burglaries, what is the probability that Jones and Elmer worked the same beats exactly three of those nights?

g. Jones is considering a new patrol strategy. He will still choose his beat randomly as before, but he will now simply stand in the center of it instead of patrolling it. If everything else remains the same (and Elmer does not change his strategy), what now is the probability of apprehension on any given night if Elmer has not previously been caught? Does your answer seem reasonable for d = 0 and d = l ?