Sample space

The sample space, usually written $S$, is the complete list of outcomes of one trial. The list has to satisfy two demands at once: the outcomes must be mutually exclusive, so no two can happen together, and exhaustive, so at least one must happen. Meet both and exactly one outcome occurs on every trial, which is why their probabilities add to 1. An event is then any subset of $S$.

A bag holds one red chip, one blue and one green. Draw a chip, record the color, put it back, and draw again, keeping the order. Stages multiply, so $S$ holds $3 \times 3 = 9$ ordered pairs and each has probability $1/9 \approx 0.1111$. The event "the two draws match" holds 3 of the 9, giving $3/9 \approx 0.3333$.

"A sample space has to be a list of equally likely outcomes." It does not. Spin a wheel divided into a half labeled A, a quarter labeled B and a quarter labeled C. Then $S = \{A, B, C\}$ with probabilities 0.5, 0.25 and 0.25: exclusive, exhaustive, adding to 1, and a perfectly good sample space. Equal likelihood is a separate assumption, and what it buys you is the right to count instead of add. The requirement that actually binds is the total, so if your probabilities do not reach 1 you have left an outcome out, and that check is the cheapest way to catch the omission before it spoils everything downstream.

A sample space need not be finite, or even listable. Waiting for a bus that arrives at some point in the next ten minutes gives $S$ as every real number from 0 to 10. Nothing can be counted there, probability comes from area under a density curve, and each exact arrival time has probability 0 while the bus still arrives.

Writing $S$ down first is the habit that makes probability problems reliable, since every rule in Unit 2 is stated over one. Once a process runs in stages, a tree diagram is the usual way to keep the list complete without dropping a branch.