The original version of this problem concerns manuscripts and proofreaders, instead of source code and programmers. It was posed and solved by George Polya and published in 1976 on The American Mathematical Monthly under the name of Probabilities in Proofreading. Because the problem is interesting and Polya’s solution is short and elegant, I have decided to record and share it. Also, since code sharing and reading is a frequent activity in the software development world, estimating the desired value can be helpful for some readers of this blog.

Estimating the number of unnoticed errors

Let E be the number of all errors, noticed and unnoticed, in the source code. Our goal is to estimate the value of E-(A+B-C). Let p be the probability that friend notices any given error and q the analogous probability for friend . The expected number of errors that may be detected by is and by is . Assuming that these probabilities are independent, the expected number of errors that may be mutually detected by both friends is .

Because we are interested in an estimate, we can safely assume that the expected numbers are approximately equal to the number of errors detected, that is, , , and . (We use the notation to denote that two numbers are approximately equal.)

We now have all the ingredients to conclude the solution. Recall that our goal is to estimate the value of E-(A+B-C). We calculate:

This is the desired estimate!

Related Articles:

• Multiples in the Fibonacci series
• Calculational proofs are usually direct

Direct proofs

The concept of direct proof can vary slightly from person to person. For instance, Wikipedia defines it as:

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions.

Alternatively, in Larry W. Cusick’s website we can read:

A direct poof [sic] should be thought of as a flow of implications beginning with “P” and ending with “Q”.

P -> … -> Q

Most proofs are (and should be) direct proofs. Always try direct proof first, unless you have a good reason not to.

I consider the wording ‘without making any further assumptions‘ in the first definition ambiguous and I don’t understand why the second definition only applies to implications. But anyway, with these definitions in mind, a direct proof for the irrationality of can be something like:

Or, alternatively, we can also use a proof of the following shape:

Irrational numbers

An irrational number is a real number that can’t be expressed as a simple fraction. Therefore, the number is irrational because for all integers m and n, with n non-negative, we have that:

A direct proof for the irrationality of

Now that we have clarified the concepts above, we prove that is irrational. For all integers m and n, with n non-negative, we have:

Note that, unlike traditional proofs, we don’t assume that m and n are co-prime, nor that is a rational. We essentially derive the boolean value of the expression

If you have any suggestions or corrections, please leave a comment. I’d be more than happy to hear from you.

Note: I learnt the contrapositive of this proof from Roland Backhouse (page 38, Program Construction — Calculating Implementations from Specifications).