Solve the following question and show ALL work please! Initially, one person kno

Question

Solve the following question and show ALL work please!

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. Let X_n denote the number of new people who learn the rumor on day n. Find P[X_2=k], k = 0,1,2,.... 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

Given,

probability of passing rumour to 1 person = 4/5

probabiltiy of passing rumour to no person= 1/5

Now,

a)for 1st day

two choices available

probability that

1 person knows the rumour= 4/5

0 persons knows the rumour = 1/5

Now, comming to second day probabilty that,

new ones knows the rumour = 4/5 * 4/5 = 16/25

p(x2=1)=16/25

0 persons knows the rumour = 1/5 (first day)+ 4/5* 1/5 (second day if person on first day dont passes the rumour)

p(x2=0)=9/25

more than 1 new persons know is not possible on one day

p(x2=2,3,...)=0

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b) If two persons knows the rumour
p(x1=0)= 1/5*1/5

p(x1=1)=2*4/5*1/5

p(x1=2)=4/5*4/5

p(x2=0)=1/5*1/5+(2*4/5*1/5)*1/5+(4/5*4/5)*1/5*1/5

=1/25*(1+2*4/5+4/5*4/5)=81/625

This p(x2=0), p(x3=0), p(x4=0) goes on decreasing

tends to

(2n+1)^2/25^(n-1)

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