50 Challenging Problems in Probability [Part 28]: Catching the Greedy Counterfeiter
Hi, I’ve recently developed an interest in problems related to probability. I came across this book “Fifty Challenging Problems in Probability with Solutions” by Frederick Mosteller. I thought it would be interesting to create a series discussing these captivating problems that might arise as interview questions. Each post will feature only 1 problem, making it a series with a total of 50 parts. Let’s dive in and activate our brain cells 🧠!
Problem:
The king’s minter boxes his coins n to a box. Each box contains m false coins. The king suspects the minter and randomly draws 1 coin from each of n boxes and has these tested.
Question: What is the chance that the sample of n coins contains exactly r false ones?
Solution:
The probability that r out of n coins drawn are the false ones is:
Let’s observe what happens when n grows larger while keeping r and m fixed. Firstly, let’s expand the Binomial coefficient:
The final expression becomes:
For large n, we have:
Therefore, the expression becomes:
which is the Poisson distribution.
And that’s all for this counterfeit coin 🟡 problem. Any feedback or questions are welcome! Check out the other problems in this series:
Thank you for reading! :)
