A sequence $M_i$ of length 100 is consisting of 50 "-1"s and 50 "+1"s randomly permuted (like balls drawn from an urn without replacing). Then a path $S_i$ is generated by the partial sums of this sequence, i.e. $S_k = \sum_{i=1}^k M_i$. The problem is to find the probability of the path not hitting +1, in other words, $\mathbb{P}(S_i \leq 0\quad \forall i=1 ... 100)$.
Suppose the path is a random walk with independent steps instead, we usually find the probability as $1 - \sum_{i=1}^{100} \mathbb{P}(\tau = i)$, where $\tau$ is the hitting time $\tau := \inf \{i \,| \,S_i = 1\}$. However, I do not know how to find $\mathbb{P}(\tau = i)$ in the problem's setting, since the probability of next step depends on all previous steps.
How could I find the distribution of this hitting time? or is there a simpler way to find this probability without thinking about the hitting time at all?
My friend told me it is easier to solve with combinatorics. It seems like what a path not hitting 1 in this problem is equivalent to what he called a Dyck path(pdf reference). Then there is some theorem says the number of Dyck path is corresponding to Catalan numbers $C_n = \frac{1}{n + 1} {2n\choose n}$. In our problem, $n=50$. That is, there exists $\frac{1}{51} {100\choose 50}$ Dyck paths. Since there are totally $100 \choose 50$ possible paths, the probability such that the path not hitting $+1$ is $\frac{1}{51} \cdot \frac {100\choose 50}{100\choose 50} = \frac1{51}$.
Please post a solution if you can find out how to do it the stochastic way.