Problem #55
Alice is having a party and has 20 guests, one of whom is her friend Bob.
Bob starts a rumor about Alice. A person hearing this rumor for the first
time will then tell the rumor to another person chosen uniformly and at
random with the exception that no one will tell the rumor to Alice or to
the person from whom they heard it. If a person who already knows the
rumor (including Bob) hears it again, they will not tell it to anyone.
- What is the probability that everyone, except Alice, will hear the
rumor before it stops propagating?
- What is the most likely number of people to hear the rumor (i.e the
mode)?
- What is the expected number of people to hear the rumor (i.e. the
mean)? A numerical approximation is fine.
- What is the median? The poster confused "mean" and "median" in the
original post.
- If 20 is replaced by n, what can be said about the asymptotic
behavior of the mode, mean, and median?