420 DISCRETE PROBABIL1T.Y a
b
C
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~,~. 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. Find a closed form for the average number of matches in that other box.
And for the number 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 t r i p h a g e : 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. 4 8 Five people stand at the vertices of a pentagon, throwing frisbees to each other.
\ f 0 -
Or, if this pentagon is in Arlington, throwing missiles at each other.
