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 , 0j<5, with probability p continues to v_i1 and with probability 1-p

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}.