A Good Robot That Is Not So Good ~ Math Crumbs

You manufacture items and notice that an item is defective with probability 0.1%.

You consider purchasing a robot to help identify defective items.

The vendor claims that the robot says that

Is this a good purchase?

Ultimately, you only care whether an item is defective if the robot says that it is defective.

Let

D

and

N

be the events that an item is defective and non-defective, and let

R_{D}

and

R_{N}

be the events that the robot says that an item is defective and non-defective.

Then, mathematically, you only care about

P (D | R_{D})

and know that

It follows (using Bayes' theorem) that

\begin{aligned} P (D | R_{D}) & = P (R_{D} | D) \frac{P (D)}{P (R_{D})} \\ = P (R_{D} | D) \frac{P (D)}{P (R_{D} | D) P (D) + P (R_{D} | N) P (N)} \\ = P (R_{D} | D) \frac{P (D)}{P (R_{D} | D) P (D) + (1 - P (R_{N} | N)) (1 - P (D))} \\ = 0.98 \frac{0.001}{0.98 \cdot 0.001 + 0.01 \cdot 0.999} \\ \approx 0.0893 . \end{aligned}

In other words, if the robot says that an item is defective, then the probability that it is defective is just under 9%.

How counter-intuitive is that?