Suppose that 30% of a company's television sets are manufactured at a plant in Portland and 70% at a plant in Houston. Of those made in Portland, 3% are defective, and of those made in Houston 9% are defective.
If you buy a defective set from the company, what is the conditional probability that it was made in Houston?
To find the conditional probability that a defective set was made in Houston, we can use Bayes' theorem. Bayes' theorem states that the probability of an event A given event B is equal to the probability of event B given event A multiplied by the probability of event A, divided by the probability of event B.
In this case, let's define event A as "the set was made in Houston" and event B as "the set is defective." We want to find the conditional probability of event A given event B.
Let's calculate the different components needed for Bayes' theorem:
1. Probability of event A: The probability of a set being made in Houston is given as 70%, or 0.7.
2. Probability of event B given event A: The probability of a set being defective, given that it was made in Houston, is 9%, or 0.09.
3. Probability of event B: To calculate the probability of a set being defective, we need to consider both Portland and Houston. The probability of a set being defective, regardless of where it was made, can be calculated by adding the probability of being defective in Portland (3% or 0.03) and the probability of being defective in Houston (9% or 0.09) weighed by their respective probabilities of manufacturing, which are 30% and 70%.
Now we can use Bayes' theorem to find the conditional probability:
Conditional Probability = (Probability of event B given event A * Probability of event A) / Probability of event B
Conditional Probability = (0.09 * 0.7) / (0.03 * 0.3 + 0.09 * 0.7)
Conditional Probability = 0.063 / (0.009 + 0.063)
Conditional Probability = 0.063 / 0.072
Conditional Probability ≈ 0.875
Therefore, the conditional probability that a defective set was made in Houston is approximately 0.875, or 87.5%.