Let T(S rightarrow BH) be the average travel time of the symmetric random walk s

Question

Let T(S rightarrow BH) be the average travel time of the symmetric random walk starting from S to a Black-Hole; see the first picture. What is T(S rightarrow BH)? Let T(S rightarrow S) be the average return time of the symmetric random walk starting from S; see the first picture. What is T(S rightarrow S)? Let Pr(S rightarrow BH_1) be the probability that starting from S, the symmetric random walk ends up in the first Black-Hole BH_1; see the second the picture. What is Pr(S rightarrow BH_1)? Alice keeps rolling two fair dice until all 6 doubles (1, 1), (2, 2), ...., (6, 6) show up. What is the average waiting time?

Explanation / Answer

problem 1: not sure that n is the size of the side of the traingle or the the distance from the center i.e s

Taking n as the distance from center S,Each vertices named as a1,a2,a3 respectively.

Let t be the time taken to move n distace.

all posible paths are

1. sa1BH1 --distance in this case 3n => 3t time similary sa2BH2,sa3bh3

total of step 1 = 18t

2. sa1a2bh2 => n +   3 n + 2n => 3t+ 3t similarly sa1a3bh3 ,sa2a1bh1,sa2a3bh3 ,sa3a1bh1,sa3a2bh2

=> total of step 2 = 18t +6 3t

3.full traingle movement sa1a2a3bh1 n+ 3n+ 3n + 2n => 3t + 2 3 t similarly other 5 possibilies where 2 sides are covered before reaching black hole => total of step 5 = 18t + 12 3 t(6 times)

average travel time = total time taken to travel through all possible paths / total number of paths possible

=( 54t+18 3t) / (3+6+6) = 5.6784t

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.