Suppose that the probability of a company supplying a defective product is $a$ and the probability that the supplied product is not defective is $b$. Before each product supplied is released for further use, it must be screened through a system which detects the potential defectiveness of the product. The probability that this process successfully detects a defective product is $x$, and the probability that it does not is $y$. What is the probability that at least one defective product will pass through the system undetected after 20 products have been screened?
i.e. $a = \text{defective product}, b = \text{non-defective product}, x = \text{screening success}, y = \text{screening fail}$
The total probability can be expressed as ${[(a + b)(x + y)]}^{20}$. However, is the probability of at least one defective product represented by $k_1 (ay) + k_2 (ay)^2 + k_{3} (ay)^3 + \ldots +k_{19}(ay)^{19} + k_{20} (ay)^{20}$ where $k_{i}~, ~i \in \mathbb{Z}$ is a coefficient of the corresponding term? I'm also interested to know if there's a simple method for computing that expression, or a more efficient method of solving this question in general.
Note: This is NOT a homework problem.