Suppose there are $m$ items in total and $t$ of them are disqualified, $t=1$%$m$, in order to improve the efficiency of quality inspection, we will select $c$ products to check instead of all of them. In order to achieve a satisfactory confidence rate $P_x$, we need to determine the specific value of $c$. The probability formula is as follows:
$$\begin{align*} P_x &= P\{x \geq 1\} \\[0.3cm] &= 1-P\{x=0\} \\[0.3cm] &= 1-\frac{m-t}{m} \cdot \frac{m-1-t}{m-1} \cdot \frac{m-2-t}{m-2} \cdots \cdots \frac{m-c+1-t}{m-c+1} \end{align*}$$
that is:
$$1-\left(\frac{m-t}{m}\right)^c \leq P_x \leq 1-\left(\frac{m-c+1-t}{m-c+1}\right)^c$$
, where x is the number of unqualified items selected.
The case I need to solve is to add a few conditions on top of that, like this: Each item has a different amount of value, which we will divide into five levels, that is, $i \in [1,5]$, the higher the rank, the more important, and among the $m$ items, the proportion of items with rank $i$ is $a_i$, so the number of items with level $i$ is $m_i = m \cdot a_i$. The total number of goods is still $t$, where the number of nonconformities existing in each grade of goods is $m_i \cdot i$ .
At this time, when sampling, we no longer extract the commodity itself, but extract the index pointing to the commodity, and then determine the commodity by the index. The higher the value level of the commodity, the more indexes pointing to the commodity. The higher the probability that the commodity will be sampled. For example, here are four items A, B, C, D. Their value levels are 1, 2, 3, and 4 respectively. Then you have the first index pointing to item A, the second and third indexes pointing to item B, the fourth, fifth, and sixth indexes pointing to item C, and the seventh through tenth indexes pointing to item D, which are randomly drawn from the index during the spot check.
In this case, if the original probability formula is still used, the actual confidence rate may not reach a certain value (one commodity corresponds to multiple indexes, and the same commodity may be selected). How to write the probability formula at this time, and how to determine the number of indexes we should select at this time (instead of the number of commodities, We can't guarantee that all indices correspond to different products when selecting).
This is the first time to ask a question on the forum, the content may not be detailed enough, the format is not very neat, please forgive me!