Question

a company makes electronic gadgets. one out of every 40 gadgets is faulty, but the company doesnt 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 $35 for every faulty gadget because they have to repair the unit. what is the expected profit or loss for every batch of 40 gadgets sold? (round your final answer to two decimal places and select the most accurate option from below).

an expected loss of $8.14

an expected loss of $3.65

correct answer is not listed

an expected profit of $0.84

an expected profit of $25

an expected loss of $7.12

an expected profit of $1.63

an expected profit of $1.08

Explanation:

Step1: Calculate the number of working and faulty gadgets

In a batch of \(n = 40\) gadgets, the number of faulty gadgets \(f=1\), and the number of working gadgets \(w=40 - 1=39\).

Step2: Calculate the total profit from working gadgets

The profit per working gadget is \(P_{work}=2\) dollars. So the total profit from working gadgets \(T_{work}=39\times2 = 78\) dollars.

Step3: Calculate the total loss from faulty gadgets

The loss per faulty gadget is \(L=- 35\) dollars (negative because it's a loss). So the total loss from faulty gadgets \(T_{fault}=1\times(-35)=-35\) dollars.

Step4: Calculate the expected profit/loss

The expected profit/loss \(E=T_{work}+T_{fault}\). Substitute the values: \(E = 78+(-35)=43\) dollars. Wait, no, let's use the probability approach.
The probability of a gadget being working \(p_{work}=\frac{39}{40}\), and the probability of a gadget being faulty \(p_{fault}=\frac{1}{40}\).
The expected value formula \(E(X)=x_1p_1 + x_2p_2\), where \(x_1 = 2\) (profit for working), \(x_2=-35\) (loss for faulty).
\(E(X)=2\times\frac{39}{40}+(-35)\times\frac{1}{40}=\frac{78 - 35}{40}=\frac{43}{40}=1.075\approx1.08\)

Answer:

An expected profit of \(\$1.08\)