Question

A large company has two production bases in the north and the south. In 2004, the output of a certain product was 200,000 pieces, 60% of which were produced in the south base. The rate of non-genuine products in the south base was 0.1%, and the rate of genuine products in the north base was 99.6%. , ask, what is the probability that a defective product found in the market is produced by the northern base?

Answer 1

To find the probability that a defective product found in the market is produced by the northern base, we use conditional probability and Bayes' theorem. The probability is approximately 0.73.

Step-by-step explanation:

To find the probability that a defective product found in the market is produced by the northern base, we need to use conditional probability. Let's define the events:

• N: A product is produced by the northern base.
• D: The product is defective.

We want to find P(N|D), which represents the probability that a product is produced by the northern base given that it is defective.

Using Bayes' theorem, we can calculate this probability:

P(N|D) = (P(D|N) * P(N)) / P(D)

P(D|N) = 1 - 0.996 = 0.004 (the rate of non-genuine products in the north base)

P(N) = 1 - 0.6 = 0.4 (the proportion of products produced in the north base)

P(D) = (P(D|N) * P(N)) + (P(D|S) * P(S)) = (0.004 * 0.4) + (0.001 * 0.6) = 0.0016 + 0.0006 = 0.0022

Now, substituting these values into Bayes' theorem:

P(N|D) = (0.004 * 0.4) / 0.0022 = 0.0016 / 0.0022 = 0.727 ≈ 0.73