Sample Space vs Event

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.

An event is a subset of the sample space, named with a capital letter such as $A$ or $B$. An event holding a single outcome is called simple, one holding several is compound. Add up the probabilities of every outcome it contains and you have $P(A)$, so when those outcomes are equally likely the adding collapses into counting.

Roll one fair die. Let $A$ be "the roll is even", the set $\{2, 4, 6\}$, and let $B$ be "the roll is greater than 3", the set $\{4, 5, 6\}$. Then $P(A) = 3/6 = 0.5$ and $P(B) = 3/6 = 0.5$. Events combine into new events: "A and B" is $\{4, 6\}$ with probability $2/6 \approx 0.3333$, and "A or B" is $\{2, 4, 5, 6\}$ with probability $4/6 \approx 0.6667$.

"A or B means one of them but not both." Read that way, "A or B" would be $\{2, 5\}$ with probability $2/6 \approx 0.3333$, which is wrong. In probability "or" is inclusive: the event holds whenever at least one of $A$ and $B$ holds, so 4 and 6 stay in. The mirror-image slip is to add, $0.5 + 0.5 = 1$, which claims every roll satisfies "A or B" when a 1 or a 3 satisfies neither. The addition rule exists to strip out the outcomes counted twice.

Not every event can be handled by listing. "The mean of 40 measurements exceeds 12" is an event on a sample space far too large to write down, and its probability comes from a model rather than a count. At the two extremes, the empty set and the whole sample space are both events, with probabilities 0 and 1.

Events are what the rest of Unit 2 operates on. The complement of an event, the union of two events, and their intersection are all events themselves, which is what lets the rules be chained together.