sample space

The sample space is the complete set of all possible outcomes of a random experiment — the full menu, written out, of everything that could happen. It is usually written with the Greek capital letter Omega, or sometimes S. For one coin toss the sample space is {heads, tails}; for one die it is {1, 2, 3, 4, 5, 6}; for the time you wait for a bus it might be every nonnegative number of minutes, an infinite continuous range. Listing this set is the very first thing you do when you set up a probability problem, because everything afterward is measured against it.

Getting the sample space right means it must be exhaustive and its members must be mutually exclusive: every actual result of the experiment is exactly one element of the set, never two at once, and never something left off the list. For two coin tosses, the careful sample space is {HH, HT, TH, TT}, not {two heads, one head, no heads} unless you specifically only care about counts — because the four ordered results are the genuinely distinct things that can happen. A sloppy sample space quietly breaks the arithmetic later, so it pays to picture exactly one run of the experiment and ask what single result it produces.

Sample spaces come in flavours. A finite one has a countable, listable number of outcomes (the six die faces). A countably infinite one goes on forever but can still be enumerated (the number of coin tosses until the first heads: 1, 2, 3, ...). A continuous one fills an interval or region (a waiting time, a dart landing on a board). Which flavour you have decides which tools you reach for next, but in every case the sample space is the universe inside which all probabilities live.