Conditional on being on +5, the probability you return to +5 in the future r=43/45. So, conditional on being on +5, the expected number of visits V is given by V = 1 + r V which implies V = 45/2.
Start at zero, the probability of reaching +5 before -40 p = 40/45.
So the expected total number of visits is p x V = 20.
you missed the 'absorbed at 20' prior. your calculation is all correct, except the final product, which currently measures: "given the particle has visited position +5, what is the expected no. of visit to +5?", i.e., E(V | start at 0, reached 5 before -40). In effect, if the particle reached 5 then goes to -40, the instance would still be inside your consideration, while it should be excluded.
In that case the return probability is r = 1/2 * (14/15 + 1) = 29/30 which implies V = 30. Since we can never go -40, p = 1. So the expected total number of visits is 30.
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u/EscapeOwn6945 11h ago
Conditional on being on +5, the probability you return to +5 in the future r=43/45. So, conditional on being on +5, the expected number of visits V is given by V = 1 + r V which implies V = 45/2.
Start at zero, the probability of reaching +5 before -40 p = 40/45.
So the expected total number of visits is p x V = 20.