Question

1. Ron Graham is a prolific author of mathematical papers. His friend, Don Knuth, reads

all of Ron’s papers and realizes that, on average, there are 4 typos for every 100 page of writing.

(a) Ron writes a new paper, that is 20 pages long. Don, before reading the actual paper,

would like to anticipate the probability that the first half of the paper has no typos, using an

exponential random variable. Which exponential r.v. would Don use? What is the probability

Don calculates?

(b) What is the expected page number for the second typo?

Answer 1

a) The mean number of typos every page is 4/100 as there are 4 typos every 100 pages. Therefore mean typos per page is given here as: 4/100 = 0.04

The mean number of typos in the first half of paper is computed here as:
= 0.5*20*0.04 = 0.4

Therefore the random variable for waiting time for a type is obtained here as:

The probability that the first half has no typos is computed here as:

Therefore 0.6703 is the required probability here.

b) The expected page number for the second typo is computed here as:

= 2 / p where p is the probability of finding an error on any page

= 2 / (1 - e^-0.04) = 51

Therefore 51 is the expected page number here.