\section*{Problem 6}
A product factory manufactures 2 types of products: product-I and product-II. Part-A is necessary for product-I, and both part-$A$ and part-$B$ are necessary for product-II. There are parts that have standard quality and parts that do not have standard quality among part-$A$ and part-$B$. All parts are delivered from the part factory to the product factory, but there is no quality check of any part. The qualities of part-$A$ and part-$B$ are independent, and they will not affect each other. The probabilities that part-$A$ and part-$B$ have standard quality are $a$ and $b$, respectively.

A final quality inspection is made in the product factory for product-I and for product-II before shipment. The inspection judges whether the quality of each product meets the standard or not. The inspections will not affect each other. The product inspection is not perfect: namely, products that have standard quality pass the product inspection as acceptable with the probability $x$. The products that do not have standard quality pass the product inspection as acceptable with the probability $y$.

Answer the following questions:\\
I. A product-I is randomly sampled and inspected once. Here, the probability that product-I can be manufactured with standard quality is defined as follows:

\begin{itemize}
  \item The probability that product-I has standard quality is $c$ if part-$A$ has standard quality.
  \item Product-I will never have standard quality if part-A does not have standard quality.
\end{itemize}

\begin{enumerate}
  \item Show the probability that the selected product-I passes the product inspection as acceptable.
  \item Show the probability that the selected product-I actually has standard quality after it has passed the product inspection as acceptable.
\end{enumerate}

II. A product-II is randomly sampled and inspected $n$ times. Here, the probability that product-II can be manufactured with standard quality is defined as follows:

\begin{itemize}
  \item The probability that product-II has standard quality is $c$ if both part-$A$ and part-$B$ have standard quality.
  \item The probability that product-II has standard quality is $d$ if only either part-$A$ or part-$B$ has standard quality.
  \item Product-II will never have standard quality if both part-$A$ and part-$B$ do not have standard quality.
\end{itemize}

\begin{enumerate}
  \item Show the probability that the selected product-II has standard quality.
  \item Show the probability that the selected product-II actually has standard quality after it has passed all product inspections (i.e., $n$ times) as acceptable.
\end{enumerate}