Estimating Unseen Bugs in Software

Q: Two engineers independently do quality assurance testing a large swath of code and discover $$e_1$$ and $$e_2$$ number of bugs of which $$e_c$$ are common to both. The probability that each of them would find a bug given a large swath of code is $$p_1$$ and $$p_2$$ respectively. What is your best estimate of the number of unseen bugs in the code?

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A: This puzzle is inspired from W Feller's book on introduction to probability. The total number of unique bugs identified are $$e_1 + e_2 - e_c$$. Let $$B_0$$ represent the total number of bugs in the software application. We could make the following statements
$$e_1 = p_1 \times B_0 \\ e_2 = p_2 \times B_0 \\ e_c = p_1 p_2 \times B_0 = \frac{e_1 e_2}{B_0}$$
The unseen bugs are simply
$$\text{Unseen Bugs} = B_0 - (e_1 + e_2 - e_c)$$
Combining the above two equations yields
$$\text{Unseen Bugs} = \frac{e_1 e_2}{e_c} - (e_1 + e_2 - e_c)$$
which simplifies to
$$\text{Unseen Bugs} = \frac{(e_1 - e_c)(e_2 - e_c)}{e_c}$$
Notice, the final result is independent of $$B_0$$. Obviously, this may not be accurate. Assume both engineers found exactly the same bugs, i.e. $$e_1 = e_2 = e_c$$, then the number of unseen bugs would become 0 which need not always be true. Also, the above equation is undefined when $$e_c = 0$$. Nevertheless, this does provide a good way to estimate the number of unseen bugs in software. The original example done by Polya & Feller were on proof readers reading text and spotting spell errors.

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