Initially, one person knows a rumor. Suppose that a person who knows a rumor wil
ID: 3134282 • 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
On I day one person knows the rumour
II day one person knows with p = 1/5 and two persons with p = 4/5
III day one person only will know if he doesnot tell any one i.e. p = 1/25
One more person will know if he reveals the previous day or that day = 16/25
Like this it goes like a chain
P(Xn =1) = 1/5^n
P(Xn=2) = nC1 (4/5) (1/5)n-1
Thus xn is binomial with n, 4/5
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If two persons know the rumour to get the rumour extinguished both should not spread
HEnce prob for a day = 1/5 (1/5) = 1/25 (as each is independent)
For n days prob = 1/25^n