Question

you want to purchase a dvd drive for your laptop computer. assume that 75% of the drives are made outside of the united states. of the u.s.-made drives, 4% are defective; of the foreign - made drives, 5% are defective. if your drive is defective, find the probability that it is foreign made.
the probability is
(do not round until final answer. round to three decimal places as needed.)

Explanation:

Step1: Define events and probabilities

Let \(F\) be the event that the drive is foreign - made and \(D\) be the event that the drive is defective.
We know that \(P(F)=0.75\), so \(P(\text{not }F)=1 - 0.75 = 0.25\).
\(P(D|\text{not }F)=0.04\) (probability of defective given it's U.S. - made) and \(P(D|F)=0.05\) (probability of defective given it's foreign - made).

Step2: Use the law of total probability

The law of total probability states that \(P(D)=P(D|F)P(F)+P(D|\text{not }F)P(\text{not }F)\).
Substitute the values: \(P(D)=(0.05\times0.75)+(0.04\times0.25)\).

$$
LATEXBLOCK0
$$

Step3: Use Bayes' theorem

Bayes' theorem states that \(P(F|D)=\frac{P(D|F)P(F)}{P(D)}\).
Substitute \(P(D|F) = 0.05\), \(P(F)=0.75\) and \(P(D)=0.0475\) into the formula:

$$
LATEXBLOCK1
$$

Answer:

\(0.789\)