A factory manufactures an electronic component. Two production lines are used. Production line A produces $40\%$ of the components and production line B produces the rest. Some of the manufactured components have a defect that prevents them from operating at the speed specified by the manufacturer. At the output of line A, $20\%$ of the components have this defect while at the output of line B, only $5\%$ do. A component manufactured in this factory is chosen at random. We denote: A the event ``the component comes from line A'', $B$ the event ``the component comes from line B'', S the event ``the component is defect-free''.

1. Show that the probability of event $S$ is $P(S) = 0.89$.
2. Given that the component has no defect, determine the probability that it comes from line A. The result should be given to the nearest $10^{-2}$.

The municipality of a large city has a stock of DVDs that it offers for rental to users of the various media libraries in this city. Among the DVDs removed, some are defective, others are not. Among the $6\%$ of defective DVDs in the entire stock, $98\%$ are removed. It is also admitted that among the non-defective DVDs, $92\%$ are kept in stock; the others are removed.
A DVD is chosen at random from the municipality's stock. Consider the following events:

• $D$: ``the DVD is defective'';
• $R$: ``the DVD is removed from stock''.

We denote by $\bar{D}$ and $\bar{R}$ the complementary events of events $D$ and $R$ respectively.

1. Prove that the probability of event $R$ is 0.134.
2. A charitable association contacts the municipality with the aim of recovering all DVDs that are removed from stock. A city official then claims that among these removed DVDs, more than half are composed of defective DVDs. Is this claim true?