If you are looking at the distribution of a maximum, then have you considered the order statistic?
Suppose there are 1 million parts which have 1% defective parts i.e 1 million parts have 10000 defective parts. Now suppose we are taking different sample sizes from 1 million like 10%, 30%, 50%, 70%, 90% of 1 million parts and we need to calculate the probability of finding maximum 5000 defective parts from these sample sizes. As 1 million parts has 1% defective parts so value of success p is 0.01 and failure q is 0.99. Now suppose we are adding 100,000 parts in one million parts which makes total 1,000,000 parts but this newly added 100,000 parts do not have any defective parts. So now what will be the value of success p in total 1,000,000 parts to find 5000 defective parts? plz also give justification for choosing value of p?
The order statistic looks at a range of distribution including the maximum of a sample:
Order statistic - Wikipedia, the free encyclopedia
If all parts are independent of each other, then you just treat the new values as an independent random samples.
In independence, you have P(A|B) = P(A) which is what the above means mathematically.
Also if everything is an I.I.D. sample then it means that if you have x defects and y successes then the probability of defects will be x/(x+y) and if you are using a Normal distribution assumption then x/(x+y) will be in both tails.