Amy, a computer programmer, produces code that, on average, has $1$ syntax error per $165$ lines of code. Her number of syntax errors is a Poisson random variable.

$($a$)$ Amy writes a program that has $475$ lines of code. Let $x$ be the number of syntax errors in her program. Find the expected value of $x.$

$($b$)$ Suppose Amy has a program with exactly $475$ lines of code. Find the probability that this program will have more than $2$ syntax errors.

$($c$)$ Suppose Amy has $10$ programs, each with $475$ lines of code. Someone is reviewing these $10$ programs. Their goal is to find $2$ programs that each have more than $2$

syntax errors. What is the probability they will be able to stop after reviewing the $5$ th program?