Concrete mathematics : a foundation for computer science

420 DISCRETE PROBABIL1T.Y
a Once he found that the other box was empty too. What’s the probability
that this occurs? (For n = 1 it happens half the time and
for n = 2 it happens 3/B of the time.) To answer this part, find a
closed form for the generating function P(w, z) = t,,, pm,nwmzn,
where pm,,, is the probability that, starting with m matches in one
box and n in the other, both boxes are empty when an empty box
is first chosen. Then. find a closed form for P~,~.
b Generalizing your a.nswer to part (a), find a closed form for the
probability that exactly k matches are in the other box when an
empty one is first th.rown away.
C Find a closed form for the average number of matches in that other And for the number
box.
in the empty box.
4’7 Some physicians, collaborating with some physicists, recently discovered
a pair of microbes that reproduce in a peculiar way. The male microbe,
called a diphage, has two receptors on its surface; the female microbe,
called a triphage, has three:
diphage:3 triphage: 9 receptor: 0
When a culture of diphages and triphages is irradiated with a psi-particle,
exactly one of the receptors on one of the phages absorbs the particle;
each receptor is equally likely. If it was a diphage receptor, that diphage
changes to a triphage; if it was a triphage receptor, that triphage splits
into two diphages. Thus if an experiment starts with one diphage, the
first psi-particle changes it to a triphage, the second particle splits the
triphage into two diphages, and the third particle changes one of the
diphages to a triphage. The fourth particle hits either the diphage or
the triphage; then there are either two triphages (probability g) or three
diphages (probability i). Find a closed form for the average number
of diphages present, if we begin with a single diphage and irradiate the
culture n times with single psi-particles.
48 Five people stand at the vertices of a pentagon, throwing frisbees to each
other.
Or, if this pentagon
is in Arlington,
throwing missiles
at each other.
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