Question

question 3
4 pts
a company makes electronic gadgets. one out of every 50 gadgets is faulty, but the company doesnt know which ones are faulty until a buyer complains. suppose the company makes a $3 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. what is the expected profit or loss for every batch of 50 gadgets sold? (round your final answer to two decimal places and select the most accurate option from below).
an expected profit of $6.23
an expected loss of $3.25
an expected loss of $2.65
an expected profit of $1.34
an expected loss of $2.25
an expected profit of $0.87
correct answer is not listed
an expected profit of $5.80

Explanation:

Step1: Calculate the number of working and faulty gadgets

In a batch of \(n = 50\) gadgets, the number of faulty gadgets \(= 1\), and the number of working gadgets \(=50 - 1=49\).

Step2: Calculate the total profit from working gadgets

Profit per working gadget \(=\$3\). Total profit from working gadgets \(=49\times3=\$147\).

Step3: Calculate the total loss from faulty gadgets

Loss per faulty gadget \(=\$80\). Total loss from faulty gadgets \(=1\times80=\$80\).

Step4: Calculate the expected profit or loss

Expected value \(E=147-80=\$67\) for a batch of \(50\) gadgets. Wait, no, wrong approach. Let's use the expected - value formula.

Let \(X\) be the profit. The probability of a gadget being working \(p_1=\frac{49}{50}\) and the profit for a working gadget \(x_1 = 3\). The probability of a gadget being faulty \(p_2=\frac{1}{50}\) and the profit (loss) for a faulty gadget \(x_2=- 80\).

The expected value formula is \(E(X)=\sum_{i = 1}^{n}p_ix_i\).

\(E(X)=\frac{49}{50}\times3+\frac{1}{50}\times(-80)\)

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Answer:

An expected profit of \(\$1.34\)