A company makes electronic gadgets. They estimate that 3 out of every 100 gadgets is faulty, but the company doesn't know which ones are faulty until a buyer complains. Suppose the company makes a \( \$ 2 \) profit on the sale of any working gadget, but suffers a loss of \( \$ 80 \) for every faulty gadget because they have to repair the unit. Check whether the company can expect a profit in the long term. Find the expected profit per gadget. Round your answer to two decimal places.
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To find the expected profit per gadget, we can calculate the expected profit for both working gadgets and faulty gadgets based on their probabilities. 1. The probability of a gadget being working is \( \frac{97}{100} \) and the profit from a working gadget is \( \$2 \). 2. The probability of a gadget being faulty is \( \frac{3}{100} \), and the loss from a faulty gadget is \( -\$80 \). Now, we calculate the expected profit per gadget: \[ \text{Expected Profit} = \left(\text{Probability of working} \times \text{Profit from working}\right) + \left(\text{Probability of faulty} \times \text{Loss from faulty}\right) \] \[ \text{Expected Profit} = \left(\frac{97}{100} \times 2\right) + \left(\frac{3}{100} \times -80\right) \] \[ \text{Expected Profit} = \left(0.97 \times 2\right) + \left(0.03 \times -80\right) \] \[ \text{Expected Profit} = 1.94 - 2.4 = -0.46 \] Rounding to two decimal places, the expected profit per gadget is \( -0.46 \). This means that in the long term, the company can expect a loss of \( \$0.46 \) for each gadget produced.
