Reliability model

Question

What follows is an illustration of a problem that I am facing in real life. A fabric mades toy and for the delivery there are three different processes:

1. Process A: reception of the pieces.

2. Process B: mount of a battery on a piece.

3. Process C: glue all the pieces together.

After each process, the pieces are inspected and a probability that a toy is defective can be calculated. Let us note the probability that a toy is defective after process A,B and C $P_A,P_B$ and $P_C$ respectively. At each step of the process, the way to calculate $P_A,P_B,P_C$ might be different. The probability $P_B$ is computed from the pieces (data) coming from the process $A$ and the probability $P_C$ is calculated from the pieces (data) coming from process $B$, and, so from process $A$ too.

The goal is to find the probability $P$ that a toy is defective at the end of the process.

Here is what I have read and with which I do not agree: $$ P = \min(P_A,P_B,P_C) \ \ \ (1). $$

Defining the overall probability $P$ as in equation (1) is equivalent to say that each $P_A$, $P_B$ and $P_C$ are independent of each other and do not affect each other. However this can not be the case since the determination of $P_A,P_B,P_C$ comes from the same toys, just at different step of the process. To be more precise, I would say that $$ P = P_C \ \ (2). $$

Everything can happen between two processes and some toy could be broken before arriving at the end.

To illustrate why I do not agree with equation (1), let us think about the following example:

$P_A = 0.05$, $P_B = 0.1$ and $P_C = 0.2$. Then we have $P = 0.05$. But his does not take into account that the pieces will go through B and C.

Moreover, $P_B$ does not represent the probability of finding defective toys after going through process $B$ but represents the probability of finding defective toys after going through processes A and B. Same reasoning for $P_C$

For me, the inspections between each processes are then useless, and only the final inspection is important. Any advice, critics about my reasoning are welcome.