A bag contains 2 gold balls, $r$ red balls and $s$ silver balls where $r \times s$ is prime number and $r < s$. The total number of balls is 10.

1. Find the number of non-gold balls and write the probability

Total number of balls $=10$.

Number of gold balls $=2$.

Number of non-gold balls $=10-2=8$.

$P(\text{non-gold})=\frac{8}{10}=\frac{4}{5}$

1. Find the possible values of $r$ and $s$

There are 2 gold balls and 10 balls altogether, so
$r+s=10-2=8.$
Since $r\times s$ is a prime number, one of $r$ and $s$ must be 1. Given $r<s$, we have
$r=1,\quad s=7.$

2. Write the probability of choosing a red ball

Number of red balls $=1$.

Total number of balls $=10$.

$P(\text{red})=\frac{1}{10}.$