What is the probability that if a randomly chosen product is defective, then it came from machine X?

Question

Two machines, A and B, each independently produces a product. Machine X makes 70 percent of the product while Y makes 30 percent. From past statistics, 5 percent made by X and 6 percent made by Y are defective. Given that a product randomly chosen was defective, what is the probability it came from X?

How could I start with this?

Thanks.

Answer 1

Hint to get started:

Let $10000$ products be produced.

Find out how many of these products are expected to be defective and find out how many of the defective products are expected to come from $X$.

Answer 2

Hint: $$P=\frac{\text{defective units from X}}{\text{total defective units}}$$

Answer 3

Consider an event:

$$ D = \text{"a randomly selected product is deffetive"}. $$

Obviously, the probability of $D$ is conditional on $X$ and $Y$, i.e., you have to consider probabilities $P(D|X)$ and $P(D|Y)$ as well as probabilities $P(X)$ and $P(Y)$. You have all the information needed to calculate these probabilities:

$$ \begin{aligned} P(X) &= \\ P(D|X) &= \\ P(Y) &= \\ P(D|Y) &= \end{aligned} $$

The question you have to answer is: what is the probability $P(X|D)$?

To answer that question you have to apply a Bayes' theorem:

$$ P(X|D) = \frac{P(X)P(D|X)}{P(X)P(D|X) + P(Y)P(D|Y)}. $$