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Suppose you have two factories, A and B which produce two items at the same time with a probability of defect from A, $p_A$ and probability of defect from B, $p_B$

Suppose a total of $N_T$ items are made so $\frac{N_T}{2}$ from each. How would you compute the probability that $N_d$ items are defective?

It looks like you can consider this to be a sum of multinomial distributions as supposing $X_A$ represents the number of defects out of $\frac{N_T}{2}$ from factory A and $X_B$ represents the number of defects out of $\frac{N_T}{2}$ from factory B both are distributed according to multinomial distribution and we are interested in $P(X_A + X_B = N_d)$

I don't see any closed form solution to computing this probability without summing the probability of every combination which sums to $N_d$

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  • $\begingroup$ Given $N_T$ is large you can use normal approximation for binomial $\endgroup$ – kludg Dec 5 '19 at 7:03
  • $\begingroup$ If $p_A=p_B, X_A+X_B$ is also a binomial distribution with parameters $N_T, p_A$. $\endgroup$ – Shubham Johri Dec 5 '19 at 7:20
  • $\begingroup$ The result of the summation can be expressed using the hypergeometric function; I doubt that you'll find a more closed form for it than that. $\endgroup$ – joriki Dec 5 '19 at 23:01

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