From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?
Say the probability that he dies from where he stands right now is p. Then, he could comfortably make one step left and end his life with probability 1/3
Or he could take one step away and two step towards and boom...take two steps away and three steps toward...so on and so forth
Resulting in p= 1/3 + 2/3 * (1/3)^2 + (2/3)^2 * (1/3)^3 +....
Summing this infinite sequence gives me probability of dying as 3/7 (around 43%). I was rather puzzled when i learnt that the correct probability is 1/2. Cant figure out what are the other 7% ways for my drunken man to die which I missed above?