A company makes electronic gadgets. They estimate that § is faulty, but the company doesn't know which ones are fal complains. Suppose the company makes a \( \$ 2 \) profit on the gadget, but suffers a loss of \( \$ 80 \) for every faulty gadget be the unit. Check whether the company can expect a profit it Find the expected profit per gadget. Round your answer to two decimal places.

To determine the expected profit per gadget, let's first define some variables. Let \( p \) represent the proportion of faulty gadgets, and therefore \( 1 - p \) is the proportion of non-faulty gadgets. The profit from a non-faulty gadget is \$2, and the loss from a faulty gadget is \$80. The expected profit \( E \) can be calculated as follows: \[ E = (1 - p) \times 2 + p \times (-80) \] Simplifying this gives: \[ E = 2 - 2p - 80p = 2 - 82p \] To check whether the company can expect a profit, set \( E \) greater than 0: \[ 2 - 82p > 0 \] Solving for \( p \): \[ 82p < 2 \implies p < \frac{2}{82} \implies p < \frac{1}{41} \approx 0.02439 \] So if the proportion of faulty gadgets is less than approximately 2.44%, the company can expect a profit. Now, plugging in the values into the expected profit formula gives: 1. If \( p = 0 \), \( E = 2 - 82(0) = 2 \) (the company makes \$2 profit per gadget). 2. If \( p = 0.02439 \), \( E = 2 - 82(0.02439) = 0 \) (the company breaks even). Therefore, to summarize, the expected profit per gadget varies significantly based on the proportion of faulty goods, and the company needs to control this percentage to remain profitable. By following this approach, the gadget company can navigate profit margins with finesse!