Chapter Prelude · 12 min
What Do We Mean by Random?
Before the axioms, ask what chance is measuring
“Probability is orderly opinion and inference from data is nothing other than the revision of such opinion in the light of relevant new information.”
Before we measure chance, there is an awkward question worth asking: what is random? A tossed coin obeys mechanics. A shuffled deck obeys mechanics. Tomorrow's rain obeys physics. Yet we use probability for all three. Sometimes chance describes a mechanism; sometimes it describes what we do not know; often it does both jobs at once.
Three ways to read a probability
A frequency reading says that is about what a long run of similar trials settles toward. A degree-of-belief reading says it records how strongly the available information supports . Apropensity reading treats the physical setup itself as having a tendency to produce outcomes with certain chances. These are philosophical interpretations, not competing arithmetic systems.
So what can mathematics insist on?
Whatever probability means in a particular application, we want its arithmetic to be coherent. Impossible events should get zero mass, the whole possible world should get total mass one, and mutually exclusive possibilities should add. Later we will write the mature object as
Do not worry about yet. It is simply the collection of questions we agree are measurable. Chapter II will earn the axioms; Chapter IX will show why this extra symbol matters on infinite spaces.
The promise of this book
We will move in three languages whenever we can: a story about uncertainty, mathematics precise enough to check the story, and an experiment or simulation that lets you watch the mathematics breathe. Definitions will arrive after there is a reason to want them.