Probably Not, 2nd Edition

Appendix

Introduction

In order to “tell our story” adequately, there are several instances in the text where it is necessary to use some topics from calculus. For people who have not studied calculus, this is bad news.

The good news is, in the limited context that these calculations are needed, it is only necessary to understand what is being calculated, not how to do the calculations.

The purpose of this appendix is to introduce two concepts from calculus and explain, in the limited context of this book, what is being calculated. This is in no way a substitute for learning calculus, but it will be enough for a reader to understand what is being calculated and why the calculation is necessary. In order to actually perform all of these calculations, it's necessary to study calculus.

Continuous Distributions and Integrals

When we are dealing with a finite discrete probability distribution function (PDF), there is a list of values, x_i, and their associated probabilities, p_i. Since the list of values includes every possible occurrence (e.g. the integers from 2 to 12 for the rolling of a pair of dice), we know that since one of these occurrences must occur, the sum of all the probabilities must = 1, the certain event. In mathematical notation, we say that

If we have a list of probabilities that does not add up to 1, we scale, or normalize, all of these probabilities so that they ...