99% of machines function correctly at a factory and 1% are faulty. A robot comes in for quality control and correctly accepts and rejects machines 99% of the time and is wrong 1% of the time. The accepted machines are sold. Of the machines that are sold, what percentage are faulty?
I think the answer given is absolutely absurd and doesn't seem right at all.
I think the answer is 0.01%
Any help is appreciated!
On
As noted in the comment, If $N$ is the total number of machines produced, the number of sold machines is $$ \frac{99}{100}\cdot\frac{99}{100}N+\frac{1}{100}\frac{1}{100}N=\frac{9802}{10000}N $$ $\frac{N}{10000}$ of which are faulty, so the percentage of faulty on sold is $$ \frac{1}{9802}=0,010202..\% $$
It can be helpful to draw a diagram for this type of problem:
The green region represents functional machines that are also accepted—true positives—and the purple region represents non-functional machines that are accepted—false positives. The two red regions represent the rejected machines. The one on the right side of the diagram is non-functional machines that have been rejected correctly, while the region at the top of the diagram is the functioning machines that have been incorrectly rejected. You can see that the two regions are the same size, so half of the rejected machines are in fact functional. If you increase the fraction of functional machines overall, you will see that, counter to intuition, the rejection error rate actually increases: the top red region becomes larger than the right-hand red region and so a larger fraction of the total red area than before.