Typos

Two colleagues correct the same document for typos independently. The first one finds a total of X typos and the second finds a total of Y typos. In Z cases, they find the same typo. How many typos can we expect to remain uncorrected?

At first inspection this problem looks impossible to solve. It is understandable that the two colleagues were not able to spot all the typos, but how can we assess how many, if they were not identified?? However, the question has more information than it initially seems.

Let's call A the total number of typos (which is unknown), p the probability that the 1st person finds a typo and q the probability that the 2nd person finds a typo. Through simple statistical logic it is not difficult to see that there is a relationship between all these quantities:

X=pA ; Y=qA ; Z=pqA

It then becomes obvious that XY=ZA, or that A=XY/Z

The number of expected uncorrected typos is then E=A-(X+Y-Z) (Z needs to be deducted from the sum of X and Y to calculate the number of unique typos identified by both persons)

Replacing A by XY/Z, E can be expressed as XY/Z - (X+Y-Z), or in a more simplified format,

E=(X-Z)(Y-Z)/Z

This means that the expected number of uncorrected typos is simply:

(Number of unique typos identified by 1st person) x (Number of unique typos identified by 2nd person) / (Number of typos identified by both persons)