There are 6 cards with natural numbers 1 through 6 written on the front and 0 written on the back. These 6 cards are placed so that the natural number $k$ is visible in the $k$-th position for natural numbers $k$ not exceeding 6.
Using these 6 cards and one die, we perform the following trial:
Roll the die once. If the result is $k$, flip the card in the $k$-th position and place it back in its original position.
After repeating this trial 3 times, given that the sum of all numbers visible on the 6 cards is even, what is the probability that the die shows 1 exactly once? The probability is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

Let $( s , i , r ) \in E$ where $E = \{ ( s , i , r ) \in \mathbf{N}^3,\, s + i + r = M \}$. Conditional on the event $\left( \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right)$, what is the probability, denoted $p ( i )$, for a susceptible person to be infected during this day?
Each of the $s$ healthy persons meets, independently of the others, $K$ persons chosen at random from the $M$ persons in the total population. As soon as at least one of the meetings is with an infected person, the healthy person becomes infected the next morning.

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The conditional probability that $X \geq 6$ given $X > 3$ equals
(A) $\frac { 125 } { 216 }$
(B) $\frac { 25 } { 216 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 25 } { 36 }$

A random variable $X$ has the following probability distribution:

$X$	0	1	2	3	4
$P ( X )$	$k$	$2 k$	$4 k$	$6 k$	$8 k$

The value of $P \left( \frac { 1 < x < 4 } { x \leq 2 } \right)$ is equal to
(1) $\frac { 4 } { 7 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 4 } { 5 }$

A bag contains 100 balls numbered $1, 2, \ldots, 100$ respectively. A person randomly draws one ball from the bag, and each ball has an equal probability of being drawn. Under which of the following conditions is the conditional probability that the person draws ball number 7 the largest?
(1) The number of the ball drawn is odd (2) The number of the ball drawn is prime (3) The number of the ball drawn is a multiple of 7 (4) The number of the ball drawn is not a multiple of 5 (5) The number of the ball drawn is less than 10