Questions: 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 2800, making Digitalis a profit of 452. 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 a 12% chance that it is defective and an 88% 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 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 2800, making Digitalis a profit of 452. 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 a 12% chance that it is defective and an 88% 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.

Transcript text: 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 $\$ 2800$, making Digitalis a profit of $\$ 452$. 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 a $12 \%$ chance that it is defective and an $88 \%$ 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.

Solution

Solution Steps

Step 1: Calculate Probabilities

The probabilities of a luteA processor being defective and non-defective are given as follows: \[ P(X = 1) = p = 0.88 \quad \text{(non-defective)} \] \[ P(X = 0) = 1 - p = 0.12 \quad \text{(defective)} \]

Step 2: Determine Expected Profit and Loss

The expected profit from a non-defective processor is calculated as: \[ \text{Expected Profit} = p \cdot \text{Profit per Processor} = 0.88 \cdot 452 = 397.76 \]

The expected loss from a defective processor, which results in a full refund, is: \[ \text{Expected Loss} = (1 - p) \cdot (-\text{Refund Amount}) = 0.12 \cdot (-2800) = -336.0 \]

Step 3: Calculate Total Expected Value

The total expected value per processor is the sum of the expected profit and expected loss: \[ \text{Total Expected Value} = \text{Expected Profit} + \text{Expected Loss} = 397.76 - 336.0 = 61.76 \]

Step 4: Conclusion on Financial Outcome

Since the total expected value per processor is positive: \[ \text{Total Expected Value} > 0 \implies \text{Digitalis should expect to make money from selling the processors.} \]