Due: ${25}^{\text{@}6}$ November $2024$

$100$ marks

Question $1(40$ marks$)$

A phamaceutieal eompany produces sterile vials of liquid medication that go through multiple stages in a production line is shown in the following schematic diagram, slightly simplified, depicts the process.

The initial input into the process consists of vials, which are loaded onto the line in trays, each tray containing $1000$ vials. In a process known as filling, these vials are filled with a specific volume of liquid medication under sterile conditions. The filled vials are then capped with a rubber stopper and sealed, ensuring the medication remains sterile. Next, the vials are transferred to a labeling machine that applies labels to each vial. In practice, there are several labeling stations on a single line; in this example, there are three labeling stations, each operating independently.

After labeling, the vials are placed on pallets, each pallet holding $7140$ vials, before being sent to quality control. Some quality control lines can accept vials directly from pallets, but others require the vials to be sorted into bulk containers holding about $100,000$ vials each. To use these quality control lines, vials must be transferred from pallets to bulk containers in a process called repacking. In practice, several quality control lines inspect the vials; here, we have two linesone accepting vials directly from pallets, and the other from bulk containers. After quality control, the vials are packaged and shipped to distributors.

It is desired to estimate the probability that the process will function for one day without failing. Assume that the filling process has a probability of $0\mathrm{.}995$ of functioning successfully for one day, which we'll denote as $P(A)=0\mathrm{.}995.$ Assume the other process components have the following probabilities of functioning successfully during a one$-$day period; capping $P(B)=0\mathrm{.}99,$ labeling stations $P(C)=P(D)=P(E)=0\mathrm{.}95,$ quality control $P(F)=0\mathrm{.}90,$ repacking $P(G)=0\mathrm{.}90,$ and quality control $P(H)=0,98.$ Assume that these stages function independently. Find the probability that the entire process functions successfully for one day.