Been going through Mosteller’s 50 challenging problems in probability.

There are 100 coins in a box. In each box there’s 1 fake. 100 boxes are tested. What is the chance of going undetected? What if it’s
n
instead of 100? 
Same as above but now each box has
m
fakes. Same sampling procedure. What is the chance of getting exactlyr
ripoffs fromn
boxes andm
fakes? 
Spores produce mold colonies on gelatin plates in a lab. Each plate has an average of 3 colonies. What fraction of plates has exactly 3? What if the average is a large integer
m
? 
A salesman sells an average of 20 cakes on a round. What is the probability of selling an even number of cakes? (We assume sales follow poisson)
Answers
27 and 28 are solved sort of at the same time.
I will handle the general case of n
boxes, r
ripoffs, and m
fakes per box. The chance of getting a fake is m/n
. The chance of not getting a fake is 1m/n
. The chance of getting exactly r
ripoffs in n
boxes is binom(n, r) * (1m/n)^(nr) * (m/n)^r
.
If n
is unlimited and m
is limited, then the chance of success is infinitesimal. This is a binomial with unlimited trials and infinitesimal success probability: a Poisson distribution by similar reasoning as in this derivation of the exponential Taylor series. Then the chance of getting exactly r
ripoffs is e^(m) * m^r / r!
.
If m
is large (say also unlimited), then the chance of success is e^(m) * m^m/m!
. This is \(\frac{1}{\sqrt{2\pi m}}\).
For 29, the chance of getting exactly 3 colonies is e^3 * 3^3 / 3!
. This is Poisson because the probability of a specific patch being infected is small, thus idealized as infinitesimal. The rate is just the average, 3.
For 30, just assuming Poisson seems sus to me, but 1 rationalization is that the salesman has a lot of customers and never sells to 2 at once. Chance of sale is low per person. Chance of selling an even amount is sum of Poisson(n,rate=20)
for all even n
. This turns out to be cosh(rate)*e^(rate) = 1/2
. Satisfying and intuitive. I’m guessing to get the odd numbers instead, use sinh over cosh. If his rate was 1, then he’d have a 56\% chance of selling an even amount.