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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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