MathDB

In a virtually-made world, each citizen, which is assumed to be a point (i.e. without area) and labelled as

1, 2, ...

. To fight against a pandemic, these citizens are required to get vaccinated. After they get vaccinated, they need to be observed for a period of time. Now assume the location that the citizens get observe is a circumference with radius

\frac{1}{4}

on the plane. For the safety reason, it is required for distance between

m

-th citizen and

n

-th citizen

d_{m, n}

satisfying the following:

(m+n)d_{m, n}\geq 1

Here what we consider is the distance on the circumference i.e. the arc length of minor arc formed by two points. Then (a) Choose one of the following which fits the situation in reality. A. The location for observation can mostly have

8

citizens. B. The location for observation can have the upper limit on the number of citizens which is larger than

8

. C. The location for observation can have any number of citizens. (b) Prove your answer in (a).

In a dance party initially there are

20

girls and

22

boys in the pool and infinitely many more girls and boys waiting outside. In each round, a participant is picked uniformly at random; if a girl is picked, then she invites a boy from the pool to dance and then both of them elave the party after the dance; while if a boy is picked, then he invites a girl and a boy from the waiting line and dance together. The three of them all stay after the dance. The party is over when there are only (two) boys left in the pool.
(a) What is the probability that the party never ends?
(b) Now the organizer of this party decides to reverse the rule, namely that if a girl is picked, then she invites a boy and a girl from the waiting line to dance and the three stay after the dance; while if a boy is picked, he invites a girl from the pool to dance and both leave after the dance. Still the party is over when there are only (two) boys left in the pool. What is the expected number of rounds until the party ends?

1

Problems(2)