Basic Definitions

Before probability itself can be defined, a precise vocabulary is needed for the kind of process probability measures.

Experiment. An experiment is simply a process whose result is well defined -- something is done, and something is observed as the outcome.

Deterministic experiment. An experiment is deterministic when its outcome can be predicted with certainty under ideal conditions. If you know the exact initial state and the governing rule, the result follows with no uncertainty -- genetic determination of an offspring's traits given the parents' genotypes, or hitting a fixed target under perfectly controlled conditions, are treated as deterministic in this idealised sense.

Random (non-deterministic) experiment. Most of probability theory is built on experiments that are the opposite of deterministic. A random experiment is one where:

1. every possible outcome is known in advance,
2. no individual outcome can be predicted in advance, and
3. the experiment can be repeated under identical conditions. Rolling a die and tossing a coin are the two standing examples used throughout the chapter -- we know in advance every face a die can show, but not which one a particular roll will land on, and the roll can be repeated as many times as we like under the same conditions. Simple (elementary) event. A simple event, also called an elementary event or sample point, is the most basic possible outcome of a random experiment -- it cannot be broken down any further into simpler outcomes. Sample space. The sample space $S$ of a random experiment is the set of all its possible outcomes. Every point of $S$ is an elementary event. Illustration. (1)(i) Roll a die: $S=\{1,2,3,4,5,6\}$. (ii) Toss a coin: $S=\{H,T\}$. (2)(i) Toss a coin repeatedly UNTIL a head appears. There is no way to say in advance how many tosses will be needed, so $S=\{H, TH, TTH, TTTH,\ldots\}$ -- an infinite set. (ii) The number of passengers waiting at a ticket counter has sample space $S=\{0,1,2,\ldots\}$, also infinite. (3)(i) A number chosen at random between 0 and 1 has sample space $S=\{x:0<x<1\}$. (ii) The life length $t$ (in hours) of a tube light has sample space $S=\{t:0<t<1000\}$. …