We are given that a batch of products is produced by three different factories. The proportion of products from Factory 1 is 30%, from Factory 2 is 50%, and from Factory 3 is 20%. The defect rates for products from these three factories are 2%, 1%, and 1%, respectively. We want to find the probability that a randomly selected product from this batch is defective.

1. Problem Description

We are given that a batch of products is produced by three different factories. The proportion of products from Factory 1 is 30%, from Factory 2 is 50%, and from Factory 3 is 20%. The defect rates for products from these three factories are 2%, 1%, and 1%, respectively. We want to find the probability that a randomly selected product from this batch is defective.

2. Solution Steps

Let $F_1$, $F_2$, and $F_3$ represent the events that a product is from Factory 1, Factory 2, and Factory 3, respectively. Let $D$ be the event that a product is defective.

We are given the following probabilities:

$P(F_1) = 0.30$

$P(F_2) = 0.50$

$P(F_3) = 0.20$

$P(D|F_1) = 0.02$

$P(D|F_2) = 0.01$

$P(D|F_3) = 0.01$

We want to find $P(D)$, the probability that a randomly selected product is defective. We can use the law of total probability:

$P(D) = P(D|F_1)P(F_1) + P(D|F_2)P(F_2) + P(D|F_3)P(F_3)$

Substituting the given values:

$P(D) = (0.02)(0.30) + (0.01)(0.50) + (0.01)(0.20)$

$P(D) = 0.006 + 0.005 + 0.002$

$P(D) = 0.013$

3. Final Answer

The probability that a randomly selected product from this batch is defective is 0.013.