Initially, one person knows a rumor. Suppose that a person who knows a rumor wil
ID: 3133776 • Letter: I
Question
Initially, one person knows a rumor. Suppose that a person who knows a rumor will pass it on to exactly one person who doesn't with probability 4/5 and to no one with probability 1/5. However a person who knows a rumor will pass on the rumor only the day after he or she learns it. (a) Let Xn denote the number of new people who learn the rumor on day n. Find PCX2 k], k 0,1,2, (b) Now suppose that two people initially know the rumor instead of one person. Find the probability q of eventual extinction, i.e., that at some point no one further learns the rumor.Explanation / Answer
(a) On day 0, only one person knows the rumor.
Now, X2 signifies that it is day 2 and k is the number of new people who know the rumor on that day.
So, for day 1 that one person tells the rumor to one more new person with a probability of 4/5 OR does not tell to anyone with a probability of 1/5. If he tells it to one more new person on day 1, then after day 1, two people (the day zero one and the new person) know the rumor or else, just one
On day 2, if 2 people know the rumor, they can tell it to two more new persons with a probability of (4/5)2 each OR 1 more new person with a probability of (4/5) *(1/5) OR to no more person on that day with a probability of (1/5)2
Thus, generalized form will be : P[X2=k] = (4/5)k (1/5)n-k
(b) Probability q of eventual extinction = Probability that they never tell to anyone + Probability that they tell to one more person and then those 3 tell it never to anyone else + and son on...
=> q = [(1/5)k]n where k = number of persons knowing the rumor and n = number of days.