There is a basket containing at least 10 white balls and at least 10 black balls, and an empty bag. Using one die, the following trial is performed. Roll the die once. If the result is 5 or more, put 2 white balls from the basket into the bag. If the result is 4 or less, put 1 black ball from the basket into the bag. When the above trial is repeated 5 times, let $a _ { n }$ and $b _ { n }$ denote the number of white balls and black balls in the bag after the $n$-th trial ($1 \leq n \leq 5$) respectively. Given that $a _ { 5 } + b _ { 5 } \geq 7$, what is the probability that there exists a natural number $k$ ($1 \leq k \leq 5$) such that $a _ { k } = b _ { k }$? If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
There is a basket containing at least 10 white balls and at least 10 black balls, and an empty bag. Using one die, the following trial is performed.
Roll the die once. If the result is 5 or more, put 2 white balls from the basket into the bag. If the result is 4 or less, put 1 black ball from the basket into the bag.
When the above trial is repeated 5 times, let $a _ { n }$ and $b _ { n }$ denote the number of white balls and black balls in the bag after the $n$-th trial ($1 \leq n \leq 5$) respectively. Given that $a _ { 5 } + b _ { 5 } \geq 7$, what is the probability that there exists a natural number $k$ ($1 \leq k \leq 5$) such that $a _ { k } = b _ { k }$? If this probability is $\frac { q } { p }$, find the value of $p + q$.
(Here, $p$ and $q$ are coprime natural numbers.) [4 points]