In which our heroes encounter many choices, some of which may lead them tolive more happily than others, and a precise count of their number ofoptions is calculated.

Imagine writing a program to implement a student registration system at acollege or university. When a student is registering for classes,you’ll need to be able to answer questions of the form “isAlice eligible to be added to the roster for Price Theory?” to decidewhether to allow her to click to add that particular course. To do so,you’ll need to know Price Theory’s prerequisites: what classesmust you have already passed before you can take Price Theory?

I often say that when you can measure what you are speaking about, andexpress it in numbers, you know something about it; but when you cannotmeasure it, when you cannot express it in numbers, your knowledge is of ameagre and unsatisfactory kind.

This chapter introduces probability, the study ofrandomness. Our focus, as will be no surprise by this point of the book, ison building a formal mathematical framework for analyzing random processes.We’ll begin with a definition of the basics of probability: defininga random process that chooses one particular outcome from aset of possibilities (any one of which occurs some fraction of the time).We’ll then analyze the likelihood that a particularevent occurs—in other words, asking whether thechosen outcome has some particular property that we care about. We thenconsider independence and dependence ofevents, and conditional probability: how, if at all, doesknowing that the randomly chosen outcome has one particular property changeour calculation of the probability that it has a different property?

Computer scientists are speed demons. When we are confronted by acomputational problem that we need to solve, we want to solve that problemas quickly as possible. That “need for speed” has driven muchof the advancement in computation over the last 50 years. We discover fasterways of solving important problems: developing data structures that supportapparently instantaneous search of billions of tweets or billions of userson a social networking site; or discovering new, faster algorithms thatsolve practical problems—such as finding shorter routes for deliverydrivers or encrypting packets to be sent over the internet.

Logic is the study of truth and falsity, of theorem and proof, of validreasoning in any context. It’s also the foundation of all of computerscience, the very reasoning that you use when you write the condition of anif statement in a Java program, or when you design an algorithm to beat agrandmaster at chess. More concretely, logic is also the foundation of allcomputers. At its heart, a computer is a collection ofcarefully arranged wires that transport electrons (which serve as a physicalmanifestation of information) and “gates” (which serve asphysical manifestations of logical operations to manipulate thoseelectrons).

This book has introduced the mathematical foundations of computerscience—the conceptual building blocks of, among other things, thelarge, complex computational systems that have become central aspects of ourdaily lives, some of which have already genuinely and meaningfully improvedthe world in their own unique ways, profound and small. Understanding andreasoning about these fundamental building blocks is necessary for you tounderstand, develop, and evaluate the key ideas of these many newapplications of computer science, and introducing these foundations has beenthe underlying goal of this book.

In computer science, a graph means a network: a collectionof things (people, web pages, subway stations, animal species, . . .) wheresome pairs of those things are joined by some kind of pairwise relationship(spent more than 15 minutes inside an enclosed space with, has a [hyper]linkto, is the stop before/after on some subway line, is a predator of, . . .).It’s possible to make graphs sound hopelessly abstract and utterlyuninteresting—a graph is a pair〈V, E〉, where V is a nonemptycollection of entities called nodes and E is acollection of edges that join pairs ofnodes—but graphs are fascinating whenever the entities andthe relationship represented by the edges are themselves interesting!Here are just a few examples.

Recursion is a powerful technique in computer science. If wecan express a solution to problem X in terms of solutionsto smaller instances of the same problem X—and wecan solve X directly for the “smallest”inputs—then we can solve X for all inputs. There aremany examples. We can sort an array A by sorting the lefthalf of A and the right half of A andmerging the results together; 1-element arrays are by definition alreadysorted. (That’s merge sort.)