The question is $\mathbb{P}$($X_{i}$ =1)=$\mathbb{P}$($X_{i}$ = -1)=1/2 for all 1≤i≤2n. For every 1≤i≤2n, Define $S_{i}$=$X_{1}$+$X_{2}$+...$X_{i}$. What is the conditional probability $\mathbb{P}$($S_{i}$≥0 for all 1≤i≤2n | $S_{2n}$=0)?
And my thought is
$\mathbb{P}$($S_{i}$≥0 for all 1≤i≤2n | $S_{2n}$=0)=$\frac{\mathbb{P}(S_{i}≥0 for all 1≤i≤2n ,S_{2n}=0)}{\mathbb{P}(S_{2n}=0)}$= $\frac{\mathbb{P}(S_{2n}=0|S_{i}≥0 for all 1≤i≤2n)*\mathbb{P}(S_{i}≥0 for all 1≤i≤2n)}{\mathbb{P}(S_{2n}=0)}$=$\frac{\mathbb{P}(S_{2n}=0|S_{2n-1}≥0)*\mathbb{P}(S_{i}≥0 for all 1≤i≤2n)}{\mathbb{P}(S_{2n}=0)}$
And my problem is: I don't know wheather [$\mathbb{P}(S_{2n}=0|S_{i}≥0 for all 1≤i≤2n)$=$\mathbb{P}(S_{2n}=0|S_{2n-1}≥0)$] is correct? Also I don't know how to solve $\mathbb{P}(S_{i}≥0 for all 1≤i≤2n)$.