You are walking along the x axis and you start at . At each time step you move to with probability and to with probability . What is the expected number of steps until you return to 0 for the first time?
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Should I post a solution or are people trying this? o.o
I looked at it for 10 min and decided it was more than I wanted to spend time on.
number of steps 1 with probability 0
number of steps 2 with probability 1/2!
number of steps <2n-1> with probability 0
number of steps <2n> with probability 1/(n+1)!
Does this make sense?
If n is number of steps then
distance = (1+x)[n/(1+x)] + (1-x)[nx/(1+x)] = 0
gives n(1+x^2) = 0
n = 0 since 1+x^2 is not zero
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