\begin{tabular}{l} Digitalis is a technology company that makes high-end computer processors. Their newest processor, the luteA, is going to be sold directly to the public. The \\ processor is to be sold for \( \$ 4500 \), making Digitalis a profit of \( \$ 383 \). Unfortunately there was a manufacturing flaw, and some of these luteA processors are \\ defective and cannot be repaired. On these defective processors, Digitalis is going to give the customer a full refund. Suppose that for each luteA there is an \\ \( 11 \% \) chance that it is defective and an \( 89 \% \) chance that it is not defective. \\ If Digitalis knows it will sell many of these processors, should it expect to make or lose money \\ from selling them? How much? \\ To answer, take into account the profit earned on each processor and the expected value of the \\ amount refunded due to the processor being defective. \\ Digitalis can expect to make money from selling these processors. \\ In the long run, they should expect to make \( \square \) dollars on each processor sold. \\ Digitalis can expect to lose money from selling these processors. \\ In the long run, they should expect to lose \( \square \) dollars on each processor sold. \\ Digitalis should expect to neither make nor lose money from selling these processors. \\ \hline\end{tabular}
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To determine whether Digitalis will make or lose money when selling the luteA processors, we can calculate the expected value based on their profitability and the refund costs. With each processor sold, the company earns a profit of $383. However, there’s an 11% chance that a processor will be defective, which means a refund of $4500 will be given back. Thus, the expected loss from the defective processors is calculated as follows: Expected refund for defective processors = \(0.11 \times 4500 = 495\). Subtracting the expected refund from the profit gives us the net outcome per processor: Profit = \(383 - 495 = -112\). Therefore, Digitalis can expect to lose $112 on each processor sold in the long run. Now, let's spice things up! Did you know that historical companies like Apple faced massive recalls in their early days due to defective products, ultimately leading to stronger quality control measures? They learned that a solid reputation is worth more in the long run than quick profits! Also, in today’s tech landscape, companies often incorporate warranty programs to mitigate losses from defects. By offering repair or replacement options, they maintain customer loyalty while minimizing refund costs. It’s a classic case of turning a potential negative into a positive business relationship!
