Consider a game where points are awarded in $n$ independent trials. In each trial, either $+1$ or $-1$ is awarded and both outcomes have the same probability of $1/2$. Let $X _ { k }$ be the point awarded in the $k ^ { \text {th } }$ trial $( 1 \leq k \leq n )$, and $S _ { k } = \sum _ { i = 1 } ^ { k } X _ { i }$. In the following questions, $n$ is an even integer such that $n \geq 4$, and $t$ is an even integer such that $2 \leq t \leq n$.
I. Obtain the probability for $S _ { 4 } = 0$.
II. Let $P _ { n } ( t )$ be the probability for $S _ { n } = t$. Find $P _ { n } ( t )$.
III. Let $P _ { n } ^ { + } ( t )$ be the probability for $S _ { 1 } = 1$ and $S _ { n } = t$. Find $P _ { n } ^ { + } ( t )$.
IV. Let $P _ { n } ^ { - } ( t )$ be the probability for $S _ { 1 } = - 1$ and $S _ { n } = t$. Find $P _ { n } ^ { - } ( t )$.
V. Let $Q _ { n } ( t )$ be the probability that all of the variables $\left\{ S _ { j } \right\}$ $( j = 1,2 , \cdots , n - 1 )$ are greater than zero and $S _ { n } = t$. Express $Q _ { n } ( t )$ with $P _ { n } ^ { + } ( t )$ and $P _ { n } ^ { - } ( t )$. Then, express $Q _ { n } ( t )$ with $P _ { n } ( t )$.
VI. Obtain the probability that all of the variables $\left\{ S _ { j } \right\} ( j = 1,2 , \cdots , n )$ are greater than zero.