The number of typos in a book follows Pois(lambda=1 per 35 pages) distribution.?

Question

a)Suppose you proofread 3 books each of 100 pages. What is the probability you find at least 3 typos?

b)Suppose you proofread one book of 400 pages. Given that you found 2 typos in the entire book, what is the probability that they were in the first 100 pages?

c) For every book you proofread, you get paid $1.20 if you find at least 2 mistakes. Suppose you proofread 5 books of 100 pages each, what is the probability you earn $3.60?

d)You are given one book of 150 pages to proofread. If you do not find an error, you get paid $1.00, if you find exactly one error you get paid $1.50, for two mistakes $2.00, for three errors, you get paid $2.50 and $5.00 for more than 3 mistakes. What are your expected earnings?

Answer 1

Dear ianboen86,

Since there are only a few minutes left before this question closes, my answer will be a bit rushed. It's also been a while since I have worked problems with the Poisson distribution, so I'll do my best off the top of my head, since time doesn't permit me to review or be more thorough.

Let f(k;l) = l^k exp(-l) / k! be the Poisson distribution,
where k is a nonnegative integer that can represent some number of "successes," and
l is a parameter for the distribution, often thought of as the rate of successes.

a) Here l = 300(1/35) = 300 / 35 = 60 / 7.
P(at least three typos) = 1 - [f(0;l) + f(1;l) + f(2;l)]
= 1 - 0.02866
= 0.97134 (to five decimal places).

b) Here l1 = 100 / 35, l2 = 300 / 35, l3 = 400 / 35.
P(2 typos in the first 100 pages | 2 typos in the first 400 pages)
= P(2 typos in the first 100 pages & 2 typos in the first 400 pages) / P(2 typos in the first 400 pages)
= P(2 typos in first 100 pages & 0 typos in the next 300 pages) / P(2 typos in the first 300 pages)
= P(2 typos in the first 100 pages) P(0 typos in the next 300 pages) / P(2 typos in the first 400 pages)
= f(2;l1) f(0;l2) / f(2;l3)
= (0.23442) (0.00019) / 0.00071
= 0.00005 (to 5 decimal places).

c) l = 100 / 35.
P(at least 2 typos in one book) = 1 - [f(0;l) + f(1;l)]
= 0.54406 (to five decimal places).
Now use the binomial distribution to find the probability of finding 2 typos in 3 of the five books. It's unclear whether you would want to know the prbability of earninge exactly $3.60 or at least $3.60, so I calculate both.
P(earn $3.60 exactly) = 0.33478 (to five decimal places).
P(earn at least $3.60) = 0.58218 (to five decimal places).

d) l = 150 / 35.
(0.01376) ($1.00)
+ (0.05899) ($1.50)
+ (0.12640) ($2.00)
+ (0.18057) ($2.50)
+ (0.62027) ($5.00)
= $3.91 .