# 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?

Back to the Advanced Problem Archives

Back to the Math Department Homepage.