4.0: Introduction

Last updated
Save as PDF

Page ID: 95644

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Probability is the study of uncertainty. This may seem like a hopeless endeavor, sort of like knowing the unknowable, but it’s not. The study of probability gives us tools for taming the uncertain world we live and program in, and for reasoning about it in a precise and helpful way.

We may not know exactly how long a particular visitor is willing to wait for our webpage to load in their browser, but we can use probability to estimate how much traffic we’ll lose if this takes longer than a certain average duration. We may not know which specific passwords a hacker will try as he attempts to break our security protocol, but we can use probability to estimate how feasible this approach will be for him. We may not know exactly when a certain program will run out of RAM and have to swap its data out to virtual memory, but we can predict how often this is likely to occur — and how painful it will be for us — given a certain system load and user behavior.

The trick is to use the tools we’ve already built — sets, relations, functions — to characterize and structure our notions of the relative likelihood of various outcomes. Once those underpinnings are secured, a layer of deductive reasoning will help us make good use of that information to begin to predict the future.