Suppose $(S_n)_{n\geq0}$ is a simple symmetric random walk, $$S_n = S_{n-1}+Y_n, S_0 = 0 $$ Where there are $Y_n$ independant variables, $$P(Y_n=1)=P(Y_n =-1)=\frac{1}{2}, \forall n\geq1$$
How would you calculate,
$$P(S_4 =0|S_2=0,S_1=1)$$ and $$P(S_4 =0|Y_2=1,Y_1=1)$$ For, $$P(S_4 =0|S_2=0,S_1=1) = \frac{P(S_4 =0,S_2=0,S_1=1)}{P(S_2=0,S_1=1)} = \frac{\frac{2}{16}}{\frac{1}{4}}$$ Which I believe represents the events of $P(S_4 =0,S_2=0,S_1=1)$ over the events where $P(S_2=0,S_1=1)$ But im not sure this is the case.
For $P(S_4 =0|Y_2=1,Y_1=1)$ Im not sure where to begin at all, some guidance would be helpful.
Process is Markov. $P(S_4=0|S_2=0,S_1=1)=P(S_4=0|S_2=0)=1/2$
$P(S_4=0|Y_2=1,Y_1=1)=P(S_4=0|S_2=2)=1/4$