One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4 . The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 4 } { 9 }$
(4) $\frac { 3 } { 5 }$

There are two boxes, A and B.
In box A, there are three cards on which the number 0 is written, two cards on which the number 2 is written, and one card on which the number 3 is written.
In box B, there are two cards on which the number 1 is written, and three cards on which the number 2 is written.
Take two cards together from box A and one card from box B. Denote the product of the numbers on the three cards by $X$.
The total number of values which $X$ can take is A. The maximum value which $X$ can take is $\mathbf{BC}$. The minimum value which $X$ can take is D.
The probability that $X = \mathrm{BC}$ is $\frac{\mathbf{E}}{\mathrm{F}}$, and the probability that $X = \square\mathrm{D}$ is $\frac{\mathbf{H}}{\mathbf{I}}$.

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf { T U }$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O } \mathbf { P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf{TU}$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.

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.