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 profit on the sale of any working
gadget, but suffers a loss of 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.

Question

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

profit on the sale of any working
gadget, but suffers a loss of

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.

UpStudy AI · Accepted Answer

To determine whether the company can expect a profit in the long term, we need to calculate the expected profit per gadget. We can do this by considering the probabilities of selling a working gadget and a faulty gadget, along with their respective profits and losses.

### Step 1: Define the probabilities
- Probability of a working gadget (not faulty): $$ P(W) = 1 - P(F) = 1 - \frac{3}{100} = \frac{97}{100} $$
- Probability of a faulty gadget: $$ P(F) = \frac{3}{100} $$

### Step 2: Define the profits and losses
- Profit from a working gadget: $$ P_W = 2 $$
- Loss from a faulty gadget: $$ L_F = -80 $$

### Step 3: Calculate the expected profit
The expected profit $$ E(P) $$ can be calculated using the formula:
$$
E(P) = P(W) \cdot P_W + P(F) \cdot L_F
$$

Substituting the values we have:
$$
E(P) = \left(\frac{97}{100} \cdot 2\right) + \left(\frac{3}{100} \cdot (-80)\right)
$$

Now, let's calculate this step by step.

1. Calculate the profit from working gadgets:
   $$
   \frac{97}{100} \cdot 2 = 1.94
   $$

2. Calculate the loss from faulty gadgets:
   $$
   \frac{3}{100} \cdot (-80) = -2.4
   $$

3. Combine these results to find the expected profit:
   $$
   E(P) = 1.94 - 2.4 = -0.46
   $$

### Conclusion
The expected profit per gadget is $$ -0.46 $$. This means the company can expect a loss of $$ \$0.46 $$ per gadget in the long term.

Now, I will round the answer to two decimal places:
$$
\text{Expected profit per gadget} = -0.46
$$

Thus, the company cannot expect a profit in the long term.
The company can expect a loss of \$0.46 per gadget in the long term.

Upstudy AI Solution

Answer

Solution

Step 1: Define the probabilities

Step 2: Define the profits and losses

Step 3: Calculate the expected profit

Conclusion

Beyond the Answer

Related Questions

Latest Probability Questions