Question
Please help me with these 3 questions about probability. Thank you so much Q3: Let T denote a complete binary tree of height 5. In
Please help me with these 3 questions about probability.
Thank you so much
Q3:
Let T denote a complete binary tree of height 5. In particular, T is rooted.
Suppose that a person starts walking from a vertex v on the i-th, 0
(1) Use the solution of Q2 (2) to compute the probability P(Ai). (The answer should depend only on i
i.)
(2)Compute the expected length of the persons walk before reaching either the root or a leaf for the first time.
Q4:
Let v0v1...v5 be a path.
(1) Find the function f(k) composed only of arithmetic operations, such that for every even k2, f(k) is the number of walks of length k that start at v, avoid v5, and end at their first visit of v0.
(2) Use (1) to express P(A2) from Q2 as an infinite series in the case when p=1/2
(3) Evaluate the series from (2).
(hint: Fill in the next few rows in the table of values of W(k, i),(Lecture 4) and observe that for every even k we have that W(k,2)=F(k1), where F(n) is the n-th Fibonacci number.)
Q5:
We connect 5 resistors in series, where the i-th resistor, for 0<i5, has the resistance Ri = 2^(i-1) and connect the ends to a 1V battery. Compute the electric potential V(i) at a vertex vi between the i-th and (i+1)-st resistors for i=1,2,3,4, and vertices at the ends for i=0,5 (We neglect the resistance of the wire. Choosing the flow of the electric current in either direction consistently with voltages is fine. ) Compare your answer to your answer to Q2 (2).
Here is Q2
Let v0v1..... v5 be a path. We fix an integer i, 0
and at every vertex vj , 0
1p continues to v_i+1. Let Ai be the event that the person visits v5 before v0 for the first time.
(1) Show that P(Ai)=pP(Ai1)+(1p)P(Ai+1), for 0
(2) Use (1) to compute p=1/3 for every i{1,2,3,4}.
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