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* ?