November 2, 2023

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

1. 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?

2. Same as above but now each box has m fakes. Same sampling procedure. What is the chance of getting exactly r ripoffs from n boxes and m fakes?

3. 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?

4. 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 1-m/n. The chance of getting exactly r ripoffs in n boxes is binom(n, r) * (1-m/n)^(n-r) * (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.