The problem I wanted to solve is related to this one. Imagine you're building a space ship, which is composed of many components that all have to work together for the ship to function. Also assume that you know the probability that any one of these will fail. Those probabilities are fixed, but you can create redundancies to ameliorate them. If you only have a total number of components you can distribute among these redundancies, what's the optimal distribution? An example will make it clear.
Imagine a simple case where you only have two components A and B, with probabilities of failure 1% and 4%, respectively. You can use a total of 10 components in your design, so you could use five As and 5 Bs, or one A and nine Bs, or any other distribution that totals ten. Each redundancy reduces the chance of catastrophic failure. What's the best solution, to minimize this risk?Intuitively we need more redundancy on the Bs, since they are four times as likely to croak. For this simple problem, we can see the answer by plugging everything into a spreadsheet and listing all the possibilities.
Our intuition is right. The best distribution is four As and six Bs. How do we find this in general? I worked on this problem over the course of a couple of days in my spare time. I scanned the notes and created a Prezi presentation out of it, to illustrate the interplay between
- knowledge of different techniques (analytical)
- trial and error
- making connections and exploiting them
- using examples to simplify the problem
- using examples to check work for plausibility
- seeking symmetries that make the problem easier to understand and solve