Intuition behind Gambler's Ruin Problem Solution

Question

In case of gambler's ruin (fair) the probability that a process starting at state $j$ eventually will reach state $N$ before state 0 is $\frac{j}{N}$. I understand that it should be proportional to $j$. But, intuitively, why $\frac{j}{N}$ ?. I am asking intuitive answer. Formal proof is available in any standard text book.

Answer 1

The expected value of your position is always $j$. So that means the probability of getting to $N$ times the "weight" of going to $N$ has to equal the probability of getting to 0 times the "weight" of going to 0. In symbols:

$$P(N) * (N-j) = P(0) * j$$

And since $P(0) + P(N) = 1$, we get that $P(N) = j/N$.