Summary

In this chapter, we have reviewed several background mathematical concepts that will become useful later in the book. You will find that these concepts are important for understanding the rest of this book. When studying these materials, we recommend not just remembering the “recipes” of the steps but focusing on the motivations and intuitions behind the techniques.

We would like to highlight the significance of the birthday paradox. Many of us come from an engineering background in which we were told to ensure reliability and guarantee success. We want to ensure that the product we deliver to our customers can survive even in the worst-case scenario. We tend to apply deterministic arguments such as requiring 366 people to ensure complete coverage of the 365 days. In modern data analysis, the worst-case scenario may not always be relevant because of the complexity of the problem and the cost of such a warranty. The probabilistic argument, or the average argument, is more reasonable and cost-effective, as you can see from our analysis of the birthday problem. The heart of the problem is the trade-off between how much confidence you need versus how much effort you need to expend. Suppose an event is unlikely to happen, but if it happens, it will be a disaster. In that case, you might prefer to be very conservative to ensure that such a disaster event has a low chance of happening. Industries related to risk management such as insurance and investment banking are all operating under this principle.